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Properties

Label 2-504-56.37-c1-0-37
Degree $2$
Conductor $504$
Sign $-0.991 - 0.133i$
Analytic cond. $4.02446$
Root an. cond. $2.00610$
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  + (0.938 − 1.05i)2-s + (−0.236 − 1.98i)4-s + (−1.98 − 1.14i)5-s + (−1.05 − 2.42i)7-s + (−2.32 − 1.61i)8-s + (−3.07 + 1.02i)10-s + (−3.36 + 1.94i)11-s + 3.33i·13-s + (−3.55 − 1.16i)14-s + (−3.88 + 0.939i)16-s + (0.143 + 0.248i)17-s + (2.41 + 1.39i)19-s + (−1.80 + 4.21i)20-s + (−1.10 + 5.38i)22-s + (3.26 − 5.65i)23-s + ⋯
L(s)  = 1  + (0.663 − 0.747i)2-s + (−0.118 − 0.992i)4-s + (−0.888 − 0.513i)5-s + (−0.399 − 0.916i)7-s + (−0.821 − 0.570i)8-s + (−0.973 + 0.323i)10-s + (−1.01 + 0.585i)11-s + 0.924i·13-s + (−0.950 − 0.310i)14-s + (−0.971 + 0.234i)16-s + (0.0348 + 0.0603i)17-s + (0.553 + 0.319i)19-s + (−0.404 + 0.943i)20-s + (−0.235 + 1.14i)22-s + (0.681 − 1.17i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.133i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.133i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.991 - 0.133i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ -0.991 - 0.133i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0685293 + 1.02575i\)
\(L(\frac12)\) \(\approx\) \(0.0685293 + 1.02575i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.938 + 1.05i)T \)
3 \( 1 \)
7 \( 1 + (1.05 + 2.42i)T \)
good5 \( 1 + (1.98 + 1.14i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (3.36 - 1.94i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 3.33iT - 13T^{2} \)
17 \( 1 + (-0.143 - 0.248i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.41 - 1.39i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.26 + 5.65i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 5.53iT - 29T^{2} \)
31 \( 1 + (3.72 + 6.44i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.15 + 2.97i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 3.51T + 41T^{2} \)
43 \( 1 + 11.2iT - 43T^{2} \)
47 \( 1 + (-0.0435 + 0.0753i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.11 + 3.52i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.76 + 2.17i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.20 - 3.58i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (11.2 - 6.51i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 6.18T + 71T^{2} \)
73 \( 1 + (-6.93 - 12.0i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.49 + 7.79i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 17.6iT - 83T^{2} \)
89 \( 1 + (8.59 - 14.8i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 6.46T + 97T^{2} \)
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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

−10.54385812666365246352623722240, −9.873312327304043244283916468760, −8.825357901288140859247164383006, −7.63440392645599359833717163783, −6.78472566788453309713366365066, −5.43398148598001279305501434774, −4.36970820199697333377491675315, −3.81043147742676043351856571507, −2.30669747315211360238467489247, −0.47822714304175310684561753366, 2.98911153451542801962534191967, 3.37060150712217920427017404737, 5.09160793108200486941261608212, 5.62923985249820254988169068929, 6.87526913522497992824831999685, 7.65819273961023326550188080547, 8.407893165497364196906604767011, 9.365345005527406764562908236661, 10.77604396973183117185460482546, 11.47698030388398446036351549838

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