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Properties

Label 2-1110-185.64-c1-0-35
Degree $2$
Conductor $1110$
Sign $-0.965 - 0.260i$
Analytic cond. $8.86339$
Root an. cond. $2.97714$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)2-s + (0.866 − 0.5i)3-s + (−0.499 − 0.866i)4-s + (−0.672 − 2.13i)5-s − 0.999i·6-s + (−0.0701 + 0.0404i)7-s − 0.999·8-s + (0.499 − 0.866i)9-s + (−2.18 − 0.483i)10-s − 3.93·11-s + (−0.866 − 0.499i)12-s + (−1.78 − 3.09i)13-s + 0.0809i·14-s + (−1.64 − 1.51i)15-s + (−0.5 + 0.866i)16-s + (−0.563 + 0.975i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.499 − 0.288i)3-s + (−0.249 − 0.433i)4-s + (−0.300 − 0.953i)5-s − 0.408i·6-s + (−0.0264 + 0.0152i)7-s − 0.353·8-s + (0.166 − 0.288i)9-s + (−0.690 − 0.152i)10-s − 1.18·11-s + (−0.249 − 0.144i)12-s + (−0.495 − 0.857i)13-s + 0.0216i·14-s + (−0.425 − 0.389i)15-s + (−0.125 + 0.216i)16-s + (−0.136 + 0.236i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 - 0.260i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.965 - 0.260i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.965 - 0.260i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (619, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ -0.965 - 0.260i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.259432197\)
\(L(\frac12)\) \(\approx\) \(1.259432197\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (-0.866 + 0.5i)T \)
5 \( 1 + (0.672 + 2.13i)T \)
37 \( 1 + (1.49 + 5.89i)T \)
good7 \( 1 + (0.0701 - 0.0404i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + 3.93T + 11T^{2} \)
13 \( 1 + (1.78 + 3.09i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.563 - 0.975i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.70 - 0.983i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 2.93T + 23T^{2} \)
29 \( 1 + 2.65iT - 29T^{2} \)
31 \( 1 - 6.08iT - 31T^{2} \)
41 \( 1 + (3.72 + 6.44i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 5.69T + 43T^{2} \)
47 \( 1 + 10.7iT - 47T^{2} \)
53 \( 1 + (-0.118 - 0.0685i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.46 + 1.42i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-8.34 + 4.81i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.06 + 1.76i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.49 + 6.04i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 5.98iT - 73T^{2} \)
79 \( 1 + (3.95 - 2.28i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-10.6 - 6.14i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (11.3 + 6.52i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 0.882T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.402253212188001255911423090744, −8.545539124639994957832045557900, −7.965036346260759076432621199807, −7.03434110990488793418670075699, −5.61904378328026902827257941438, −5.05576856370708622008941007843, −3.99765417754323171705786941840, −2.97136814158452911082132235368, −1.92556740622686326036218482308, −0.43515772666349057682499079011, 2.38068050637418655624050583675, 3.13832292258711447230128784615, 4.26767246138113938490107341627, 5.03884738598010832727709247539, 6.21621700058567278708872243469, 7.05004048451818294862481790917, 7.68139282245603185292574631610, 8.459357803245701836115622672766, 9.449397300203651058441328260159, 10.19413464984240436075368702596

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