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Properties

Label 1-3997-3997.790-r1-0-0
Degree $1$
Conductor $3997$
Sign $-0.525 - 0.851i$
Analytic cond. $429.537$
Root an. cond. $429.537$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.137 + 0.990i)2-s + (−0.00551 + 0.999i)3-s + (−0.962 + 0.272i)4-s + (−0.685 + 0.728i)5-s + (−0.991 + 0.131i)6-s + (−0.401 − 0.915i)8-s + (−0.999 − 0.0110i)9-s + (−0.815 − 0.578i)10-s + (0.0935 + 0.995i)11-s + (−0.266 − 0.963i)12-s + (0.360 − 0.932i)13-s + (−0.724 − 0.689i)15-s + (0.851 − 0.523i)16-s + (−0.761 − 0.648i)17-s + (−0.126 − 0.991i)18-s + (−0.583 − 0.812i)19-s + ⋯
L(s)  = 1  + (0.137 + 0.990i)2-s + (−0.00551 + 0.999i)3-s + (−0.962 + 0.272i)4-s + (−0.685 + 0.728i)5-s + (−0.991 + 0.131i)6-s + (−0.401 − 0.915i)8-s + (−0.999 − 0.0110i)9-s + (−0.815 − 0.578i)10-s + (0.0935 + 0.995i)11-s + (−0.266 − 0.963i)12-s + (0.360 − 0.932i)13-s + (−0.724 − 0.689i)15-s + (0.851 − 0.523i)16-s + (−0.761 − 0.648i)17-s + (−0.126 − 0.991i)18-s + (−0.583 − 0.812i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3997 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.525 - 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3997 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.525 - 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3997\)    =    \(7 \cdot 571\)
Sign: $-0.525 - 0.851i$
Analytic conductor: \(429.537\)
Root analytic conductor: \(429.537\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3997} (790, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3997,\ (1:\ ),\ -0.525 - 0.851i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3181820040 + 0.5702110995i\)
\(L(\frac12)\) \(\approx\) \(-0.3181820040 + 0.5702110995i\)
\(L(1)\) \(\approx\) \(0.4855589651 + 0.5928016512i\)
\(L(1)\) \(\approx\) \(0.4855589651 + 0.5928016512i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
571 \( 1 \)
good2 \( 1 + (0.137 + 0.990i)T \)
3 \( 1 + (-0.00551 + 0.999i)T \)
5 \( 1 + (-0.685 + 0.728i)T \)
11 \( 1 + (0.0935 + 0.995i)T \)
13 \( 1 + (0.360 - 0.932i)T \)
17 \( 1 + (-0.761 - 0.648i)T \)
19 \( 1 + (-0.583 - 0.812i)T \)
23 \( 1 + (-0.213 - 0.976i)T \)
29 \( 1 + (0.451 + 0.892i)T \)
31 \( 1 + (0.879 + 0.475i)T \)
37 \( 1 + (-0.256 + 0.966i)T \)
41 \( 1 + (-0.904 - 0.426i)T \)
43 \( 1 + (0.802 - 0.596i)T \)
47 \( 1 + (-0.0275 - 0.999i)T \)
53 \( 1 + (0.984 - 0.175i)T \)
59 \( 1 + (-0.945 - 0.324i)T \)
61 \( 1 + (-0.970 + 0.240i)T \)
67 \( 1 + (-0.644 - 0.764i)T \)
71 \( 1 + (0.913 + 0.406i)T \)
73 \( 1 + (0.709 + 0.705i)T \)
79 \( 1 + (-0.170 + 0.985i)T \)
83 \( 1 + (0.609 + 0.792i)T \)
89 \( 1 + (0.381 - 0.924i)T \)
97 \( 1 + (0.0385 - 0.999i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.05972977553203988215664805617, −17.25574839716422129686870865801, −16.78472998822564290276434641958, −15.867379835300779915385967250416, −14.96286049928875351613114870137, −14.10193656380032415715799056505, −13.495204197393527598269339218389, −13.11022756354954860320339654840, −12.16995082641398206452909373564, −11.85736836475138693557197149867, −11.19045292444478432833330427117, −10.583139728291184504914270138120, −9.30335342999780197539465937758, −8.870180322885872044612286025684, −8.14294121464521072175078668737, −7.68875654833539814166802128977, −6.26806880253837707903271728932, −5.998721767282037447975066910902, −4.88254083689961169597568958687, −4.06150904552407224662630254171, −3.52157766413352364882590152027, −2.47541383432925270043823255523, −1.65721328316182737723358341981, −1.0439530618970926446223636214, −0.16367086655448644272261713768, 0.59008196221107627832886066466, 2.459503069333378381474500605577, 3.16885208545891982219212216928, 3.95210700056049390976448194965, 4.69858264602279622797332163788, 5.05993158122290583865779311843, 6.241012245585201173324308862470, 6.78805203808260254603043420437, 7.45995620011634624471157775363, 8.57384909291933396851931246496, 8.638877513844672950177197441923, 9.839786966768885612685306986482, 10.36145350805443053111276914686, 11.00248177954795530710972084363, 11.99414171853598126900603942851, 12.58143772789005228154142662220, 13.66982261152568587632130367478, 14.175516191101289853256545557577, 15.05890280201239456317602044129, 15.45905394704577624699117855691, 15.63519179459928725093511634872, 16.6484218211658578194469019891, 17.24835373758982365495189330485, 18.04895616535628826780457312067, 18.37694419075034502126585905953

Graph of the $Z$-function along the critical line