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Properties

Label 2-845-65.18-c1-0-16
Degree $2$
Conductor $845$
Sign $-0.436 - 0.899i$
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  + 1.51i·2-s + (0.478 + 0.478i)3-s − 0.304·4-s + (−2.15 − 0.600i)5-s + (−0.726 + 0.726i)6-s + 2.59·7-s + 2.57i·8-s − 2.54i·9-s + (0.911 − 3.26i)10-s + (3.53 + 3.53i)11-s + (−0.145 − 0.145i)12-s + 3.93i·14-s + (−0.743 − 1.31i)15-s − 4.51·16-s + (−0.0578 − 0.0578i)17-s + 3.85·18-s + ⋯
L(s)  = 1  + 1.07i·2-s + (0.276 + 0.276i)3-s − 0.152·4-s + (−0.963 − 0.268i)5-s + (−0.296 + 0.296i)6-s + 0.980·7-s + 0.910i·8-s − 0.847i·9-s + (0.288 − 1.03i)10-s + (1.06 + 1.06i)11-s + (−0.0420 − 0.0420i)12-s + 1.05i·14-s + (−0.191 − 0.340i)15-s − 1.12·16-s + (−0.0140 − 0.0140i)17-s + 0.909·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.436 - 0.899i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.436 - 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $-0.436 - 0.899i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (408, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ -0.436 - 0.899i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.979553 + 1.56355i\)
\(L(\frac12)\) \(\approx\) \(0.979553 + 1.56355i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.15 + 0.600i)T \)
13 \( 1 \)
good2 \( 1 - 1.51iT - 2T^{2} \)
3 \( 1 + (-0.478 - 0.478i)T + 3iT^{2} \)
7 \( 1 - 2.59T + 7T^{2} \)
11 \( 1 + (-3.53 - 3.53i)T + 11iT^{2} \)
17 \( 1 + (0.0578 + 0.0578i)T + 17iT^{2} \)
19 \( 1 + (-1.98 - 1.98i)T + 19iT^{2} \)
23 \( 1 + (-2.86 + 2.86i)T - 23iT^{2} \)
29 \( 1 - 4.98iT - 29T^{2} \)
31 \( 1 + (2.32 - 2.32i)T - 31iT^{2} \)
37 \( 1 - 0.571T + 37T^{2} \)
41 \( 1 + (7.36 - 7.36i)T - 41iT^{2} \)
43 \( 1 + (-0.0967 + 0.0967i)T - 43iT^{2} \)
47 \( 1 + 2.30T + 47T^{2} \)
53 \( 1 + (-6.70 - 6.70i)T + 53iT^{2} \)
59 \( 1 + (1.89 - 1.89i)T - 59iT^{2} \)
61 \( 1 - 5.48T + 61T^{2} \)
67 \( 1 + 15.7iT - 67T^{2} \)
71 \( 1 + (5.43 - 5.43i)T - 71iT^{2} \)
73 \( 1 + 6.61iT - 73T^{2} \)
79 \( 1 + 5.71iT - 79T^{2} \)
83 \( 1 - 3.70T + 83T^{2} \)
89 \( 1 + (-12.6 + 12.6i)T - 89iT^{2} \)
97 \( 1 + 5.36iT - 97T^{2} \)
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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

−10.43301401535000463147249793089, −9.179160758557203217451470384091, −8.672790245751731446486358285517, −7.79968582532444517805412864427, −7.10356285406661360315050571369, −6.39042621185304948974475832677, −5.03567069642798759681499032789, −4.44363065814528954721006614040, −3.30405042916404198936316319819, −1.52873350429499366914582966021, 1.00365832497831718354563189982, 2.19953246795819751736659412168, 3.32549923951160777604684269584, 4.10325234139333843005471950136, 5.26035929967207282082445194107, 6.69815568293530230023609123798, 7.46673636945187264130682027870, 8.296889814810110062555519893788, 9.044959338993681400314822197007, 10.23647884671094324919050862431

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