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Properties

Label 2-845-65.28-c1-0-5
Degree $2$
Conductor $845$
Sign $-0.293 - 0.955i$
Analytic cond. $6.74735$
Root an. cond. $2.59756$
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  + (−1.94 + 1.12i)2-s + (0.514 − 1.91i)3-s + (1.53 − 2.65i)4-s + (−0.247 − 2.22i)5-s + (1.15 + 4.31i)6-s + (−0.638 + 1.10i)7-s + 2.39i·8-s + (−0.820 − 0.473i)9-s + (2.98 + 4.05i)10-s + (−1.41 + 5.27i)11-s + (−4.30 − 4.30i)12-s − 2.87i·14-s + (−4.39 − 0.666i)15-s + (0.365 + 0.633i)16-s + (−3.11 + 0.833i)17-s + 2.13·18-s + ⋯
L(s)  = 1  + (−1.37 + 0.795i)2-s + (0.296 − 1.10i)3-s + (0.766 − 1.32i)4-s + (−0.110 − 0.993i)5-s + (0.472 + 1.76i)6-s + (−0.241 + 0.418i)7-s + 0.848i·8-s + (−0.273 − 0.157i)9-s + (0.943 + 1.28i)10-s + (−0.426 + 1.59i)11-s + (−1.24 − 1.24i)12-s − 0.768i·14-s + (−1.13 − 0.172i)15-s + (0.0913 + 0.158i)16-s + (−0.754 + 0.202i)17-s + 0.502·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.293 - 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.293 - 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $-0.293 - 0.955i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (418, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ -0.293 - 0.955i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.217193 + 0.293889i\)
\(L(\frac12)\) \(\approx\) \(0.217193 + 0.293889i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.247 + 2.22i)T \)
13 \( 1 \)
good2 \( 1 + (1.94 - 1.12i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (-0.514 + 1.91i)T + (-2.59 - 1.5i)T^{2} \)
7 \( 1 + (0.638 - 1.10i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.41 - 5.27i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (3.11 - 0.833i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (1.17 - 0.315i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (0.160 + 0.0428i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (8.41 - 4.85i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.233 - 0.233i)T - 31iT^{2} \)
37 \( 1 + (-0.660 - 1.14i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.483 - 0.129i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-1.72 - 6.43i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + 3.20T + 47T^{2} \)
53 \( 1 + (-4.49 - 4.49i)T + 53iT^{2} \)
59 \( 1 + (-0.000595 - 0.00222i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (0.695 - 1.20i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.26 + 3.03i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3.14 - 11.7i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + 7.34iT - 73T^{2} \)
79 \( 1 - 11.1iT - 79T^{2} \)
83 \( 1 + 2.65T + 83T^{2} \)
89 \( 1 + (6.96 + 1.86i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (3.62 + 2.09i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.973190533523095819997104061721, −9.325731565516576870870267198800, −8.632626364842660819250316688346, −7.86445385467574321835213280987, −7.30154697578829511933577890685, −6.58458989413385604479690669943, −5.52150594868061906355404120694, −4.32633115614740930470078915050, −2.21403134661476178188291967679, −1.37713461139747705644991336144, 0.27557898599933907614561398826, 2.30251834534769965002577652905, 3.29693331470196501464841157617, 3.94188682090598800179604077578, 5.54902641259005424173502733689, 6.77847173187448542775312695256, 7.74063486849168192149127964859, 8.586643420876632945907413585915, 9.299092363968461692270821388831, 10.01251702688155489218448275327

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