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Properties

Label 2-845-169.114-c1-0-0
Degree $2$
Conductor $845$
Sign $-0.0749 - 0.997i$
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  + (−1.70 − 1.39i)2-s + (0.230 + 0.0185i)3-s + (0.571 + 2.80i)4-s + (−0.992 + 0.120i)5-s + (−0.366 − 0.352i)6-s + (0.886 + 1.40i)7-s + (0.879 − 1.67i)8-s + (−2.90 − 0.472i)9-s + (1.86 + 1.17i)10-s + (−0.460 − 2.83i)11-s + (0.0795 + 0.654i)12-s + (−0.0234 − 3.60i)13-s + (0.440 − 3.62i)14-s + (−0.230 + 0.00929i)15-s + (1.42 − 0.605i)16-s + (−0.0813 + 0.0514i)17-s + ⋯
L(s)  = 1  + (−1.20 − 0.985i)2-s + (0.132 + 0.0107i)3-s + (0.285 + 1.40i)4-s + (−0.443 + 0.0539i)5-s + (−0.149 − 0.143i)6-s + (0.334 + 0.529i)7-s + (0.311 − 0.592i)8-s + (−0.969 − 0.157i)9-s + (0.589 + 0.372i)10-s + (−0.138 − 0.854i)11-s + (0.0229 + 0.189i)12-s + (−0.00650 − 0.999i)13-s + (0.117 − 0.969i)14-s + (−0.0595 + 0.00239i)15-s + (0.355 − 0.151i)16-s + (−0.0197 + 0.0124i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0749 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0749 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $-0.0749 - 0.997i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (621, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ -0.0749 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0845357 + 0.0911246i\)
\(L(\frac12)\) \(\approx\) \(0.0845357 + 0.0911246i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.992 - 0.120i)T \)
13 \( 1 + (0.0234 + 3.60i)T \)
good2 \( 1 + (1.70 + 1.39i)T + (0.400 + 1.95i)T^{2} \)
3 \( 1 + (-0.230 - 0.0185i)T + (2.96 + 0.481i)T^{2} \)
7 \( 1 + (-0.886 - 1.40i)T + (-3.00 + 6.32i)T^{2} \)
11 \( 1 + (0.460 + 2.83i)T + (-10.4 + 3.48i)T^{2} \)
17 \( 1 + (0.0813 - 0.0514i)T + (7.28 - 15.3i)T^{2} \)
19 \( 1 + (-5.29 - 3.05i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.36 + 2.36i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.28 - 5.25i)T + (-5.80 - 28.4i)T^{2} \)
31 \( 1 + (0.548 - 2.22i)T + (-27.4 - 14.4i)T^{2} \)
37 \( 1 + (7.28 - 2.11i)T + (31.2 - 19.7i)T^{2} \)
41 \( 1 + (0.144 - 1.78i)T + (-40.4 - 6.57i)T^{2} \)
43 \( 1 + (2.01 - 6.97i)T + (-36.3 - 22.9i)T^{2} \)
47 \( 1 + (7.77 - 8.77i)T + (-5.66 - 46.6i)T^{2} \)
53 \( 1 + (5.98 + 3.14i)T + (30.1 + 43.6i)T^{2} \)
59 \( 1 + (5.10 - 11.9i)T + (-40.8 - 42.5i)T^{2} \)
61 \( 1 + (-0.00602 + 0.149i)T + (-60.8 - 4.90i)T^{2} \)
67 \( 1 + (10.2 + 2.09i)T + (61.6 + 26.2i)T^{2} \)
71 \( 1 + (6.39 - 3.03i)T + (44.9 - 54.9i)T^{2} \)
73 \( 1 + (15.0 + 5.69i)T + (54.6 + 48.4i)T^{2} \)
79 \( 1 + (-10.8 - 9.59i)T + (9.52 + 78.4i)T^{2} \)
83 \( 1 + (-10.2 + 7.05i)T + (29.4 - 77.6i)T^{2} \)
89 \( 1 + (-1.21 + 0.703i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (10.0 - 13.4i)T + (-26.9 - 93.1i)T^{2} \)
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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

−10.52702364833995061352338584440, −9.535016385961441694727146684110, −8.733632261001055901642917244581, −8.215395079680640913017811008975, −7.56816380997480805116884353603, −5.98931857167869353707610137313, −5.18905866350098962391847494884, −3.30346609990190828144741097133, −2.96488730390731008543615788394, −1.41067077218545540753251622765, 0.091905294157475917816041499195, 1.82550234585091319022309518359, 3.54276889320026353739886017832, 4.81783201853315049761077433763, 5.82671476884682197953590183614, 6.98078015858966634278368795311, 7.46038648548934387118956419724, 8.173761526056409050009496910068, 9.083342889503189833980536712824, 9.586656836245988189967281478773

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