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Properties

Label 2-806-31.4-c1-0-25
Degree $2$
Conductor $806$
Sign $0.911 + 0.412i$
Analytic cond. $6.43594$
Root an. cond. $2.53691$
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  + (0.309 + 0.951i)2-s + (0.581 − 1.78i)3-s + (−0.809 + 0.587i)4-s + 1.34·5-s + 1.88·6-s + (3.39 − 2.46i)7-s + (−0.809 − 0.587i)8-s + (−0.434 − 0.315i)9-s + (0.415 + 1.27i)10-s + (−0.504 + 0.366i)11-s + (0.581 + 1.78i)12-s + (0.309 − 0.951i)13-s + (3.39 + 2.46i)14-s + (0.781 − 2.40i)15-s + (0.309 − 0.951i)16-s + (−3.45 − 2.51i)17-s + ⋯
L(s)  = 1  + (0.218 + 0.672i)2-s + (0.335 − 1.03i)3-s + (−0.404 + 0.293i)4-s + 0.601·5-s + 0.767·6-s + (1.28 − 0.932i)7-s + (−0.286 − 0.207i)8-s + (−0.144 − 0.105i)9-s + (0.131 + 0.404i)10-s + (−0.152 + 0.110i)11-s + (0.167 + 0.516i)12-s + (0.0857 − 0.263i)13-s + (0.908 + 0.659i)14-s + (0.201 − 0.620i)15-s + (0.0772 − 0.237i)16-s + (−0.838 − 0.609i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 806 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.911 + 0.412i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 806 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.911 + 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(806\)    =    \(2 \cdot 13 \cdot 31\)
Sign: $0.911 + 0.412i$
Analytic conductor: \(6.43594\)
Root analytic conductor: \(2.53691\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{806} (469, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 806,\ (\ :1/2),\ 0.911 + 0.412i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.26430 - 0.488174i\)
\(L(\frac12)\) \(\approx\) \(2.26430 - 0.488174i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 - 0.951i)T \)
13 \( 1 + (-0.309 + 0.951i)T \)
31 \( 1 + (5.55 + 0.404i)T \)
good3 \( 1 + (-0.581 + 1.78i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 - 1.34T + 5T^{2} \)
7 \( 1 + (-3.39 + 2.46i)T + (2.16 - 6.65i)T^{2} \)
11 \( 1 + (0.504 - 0.366i)T + (3.39 - 10.4i)T^{2} \)
17 \( 1 + (3.45 + 2.51i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-1.06 - 3.26i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-4.46 - 3.24i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (0.913 + 2.81i)T + (-23.4 + 17.0i)T^{2} \)
37 \( 1 - 2.71T + 37T^{2} \)
41 \( 1 + (0.147 + 0.454i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + (1.96 + 6.04i)T + (-34.7 + 25.2i)T^{2} \)
47 \( 1 + (0.939 - 2.89i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-9.78 - 7.11i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (2.26 - 6.98i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 - 8.49T + 61T^{2} \)
67 \( 1 + 11.1T + 67T^{2} \)
71 \( 1 + (-6.69 - 4.86i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-4.60 + 3.34i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (3.32 + 2.41i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-2.18 - 6.71i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (-2.67 + 1.94i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (3.65 - 2.65i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.16820394434174723567222947639, −9.098694392862292100634552030940, −8.158955062440902758716207507020, −7.46588044913339341383777732732, −7.06203693944520893779605194368, −5.86059219220216725401663227321, −4.97643059967901528906875725761, −3.94648597437814112833314051562, −2.30231775518778495802809922214, −1.21705302077674472357372867591, 1.71712725017903791906624045508, 2.68815100727346642499770096662, 3.95394470513866951142382723723, 4.87168722342324412579641827495, 5.42763319229528669904851945343, 6.69297727180124671070993808657, 8.195366756908705639409698940453, 8.960445732872429581323825073111, 9.385591741493819235674031103868, 10.41973218436576126147742201914

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