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Properties

Label 2-812-812.503-c0-0-1
Degree $2$
Conductor $812$
Sign $-0.521 + 0.853i$
Analytic cond. $0.405240$
Root an. cond. $0.636585$
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  + (0.623 − 0.781i)2-s + (−0.222 − 0.974i)4-s + (−0.433 − 0.900i)7-s + (−0.900 − 0.433i)8-s + (−0.781 − 0.623i)9-s + (0.222 + 0.0250i)11-s + (−0.974 − 0.222i)14-s + (−0.900 + 0.433i)16-s + (−0.974 + 0.222i)18-s + (0.158 − 0.158i)22-s + (1.75 − 0.400i)23-s + (0.900 + 0.433i)25-s + (−0.781 + 0.623i)28-s + (−0.433 + 0.900i)29-s + (−0.222 + 0.974i)32-s + ⋯
L(s)  = 1  + (0.623 − 0.781i)2-s + (−0.222 − 0.974i)4-s + (−0.433 − 0.900i)7-s + (−0.900 − 0.433i)8-s + (−0.781 − 0.623i)9-s + (0.222 + 0.0250i)11-s + (−0.974 − 0.222i)14-s + (−0.900 + 0.433i)16-s + (−0.974 + 0.222i)18-s + (0.158 − 0.158i)22-s + (1.75 − 0.400i)23-s + (0.900 + 0.433i)25-s + (−0.781 + 0.623i)28-s + (−0.433 + 0.900i)29-s + (−0.222 + 0.974i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 812 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.521 + 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 812 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.521 + 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(812\)    =    \(2^{2} \cdot 7 \cdot 29\)
Sign: $-0.521 + 0.853i$
Analytic conductor: \(0.405240\)
Root analytic conductor: \(0.636585\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{812} (503, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 812,\ (\ :0),\ -0.521 + 0.853i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.149212167\)
\(L(\frac12)\) \(\approx\) \(1.149212167\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.623 + 0.781i)T \)
7 \( 1 + (0.433 + 0.900i)T \)
29 \( 1 + (0.433 - 0.900i)T \)
good3 \( 1 + (0.781 + 0.623i)T^{2} \)
5 \( 1 + (-0.900 - 0.433i)T^{2} \)
11 \( 1 + (-0.222 - 0.0250i)T + (0.974 + 0.222i)T^{2} \)
13 \( 1 + (-0.222 + 0.974i)T^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + (-0.781 + 0.623i)T^{2} \)
23 \( 1 + (-1.75 + 0.400i)T + (0.900 - 0.433i)T^{2} \)
31 \( 1 + (0.433 - 0.900i)T^{2} \)
37 \( 1 + (0.189 + 1.68i)T + (-0.974 + 0.222i)T^{2} \)
41 \( 1 + iT^{2} \)
43 \( 1 + (-1.19 - 0.752i)T + (0.433 + 0.900i)T^{2} \)
47 \( 1 + (0.974 + 0.222i)T^{2} \)
53 \( 1 + (0.277 - 1.21i)T + (-0.900 - 0.433i)T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + (-0.781 - 0.623i)T^{2} \)
67 \( 1 + (0.541 - 0.678i)T + (-0.222 - 0.974i)T^{2} \)
71 \( 1 + (-0.277 - 0.347i)T + (-0.222 + 0.974i)T^{2} \)
73 \( 1 + (-0.433 - 0.900i)T^{2} \)
79 \( 1 + (0.222 + 1.97i)T + (-0.974 + 0.222i)T^{2} \)
83 \( 1 + (0.623 + 0.781i)T^{2} \)
89 \( 1 + (-0.433 + 0.900i)T^{2} \)
97 \( 1 + (-0.781 + 0.623i)T^{2} \)
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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

−10.54996357261380118268460009511, −9.185674311701639375300079966077, −9.044339732729913238604242935263, −7.36401719895463302437793063088, −6.56642839174760492367994790459, −5.62570282726942069021743606919, −4.59469017798466834568313167181, −3.56773592863378161282438772407, −2.81271668400874590573770523835, −1.03977187993041175460159173928, 2.51936981496033461507104549681, 3.34025302566291355020819175693, 4.75809204178692826791180773183, 5.45856186950826579814149741248, 6.30225833664594189833148158733, 7.13891165424835747891618727983, 8.230676782181550812764760543735, 8.796061156247765358998787011917, 9.617335506014708386630449780734, 10.97877910620312294930708160764

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