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Properties

Label 2-845-169.114-c1-0-4
Degree $2$
Conductor $845$
Sign $0.823 - 0.567i$
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  + (−0.565 − 0.461i)2-s + (−1.03 − 0.0836i)3-s + (−0.293 − 1.43i)4-s + (−0.992 + 0.120i)5-s + (0.547 + 0.525i)6-s + (−1.93 − 3.05i)7-s + (−1.17 + 2.24i)8-s + (−1.89 − 0.307i)9-s + (0.616 + 0.389i)10-s + (0.284 + 1.75i)11-s + (0.183 + 1.51i)12-s + (−1.77 + 3.13i)13-s + (−0.318 + 2.62i)14-s + (1.03 − 0.0418i)15-s + (−1.00 + 0.426i)16-s + (−0.841 + 0.532i)17-s + ⋯
L(s)  = 1  + (−0.399 − 0.326i)2-s + (−0.598 − 0.0483i)3-s + (−0.146 − 0.718i)4-s + (−0.443 + 0.0539i)5-s + (0.223 + 0.214i)6-s + (−0.730 − 1.15i)7-s + (−0.415 + 0.792i)8-s + (−0.631 − 0.102i)9-s + (0.195 + 0.123i)10-s + (0.0859 + 0.528i)11-s + (0.0531 + 0.437i)12-s + (−0.492 + 0.870i)13-s + (−0.0850 + 0.700i)14-s + (0.268 − 0.0108i)15-s + (−0.250 + 0.106i)16-s + (−0.204 + 0.129i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.823 - 0.567i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.823 - 0.567i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $0.823 - 0.567i$
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: \((2,\ 845,\ (\ :1/2),\ 0.823 - 0.567i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.314512 + 0.0979012i\)
\(L(\frac12)\) \(\approx\) \(0.314512 + 0.0979012i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.77 - 3.13i)T \)
good2 \( 1 + (0.565 + 0.461i)T + (0.400 + 1.95i)T^{2} \)
3 \( 1 + (1.03 + 0.0836i)T + (2.96 + 0.481i)T^{2} \)
7 \( 1 + (1.93 + 3.05i)T + (-3.00 + 6.32i)T^{2} \)
11 \( 1 + (-0.284 - 1.75i)T + (-10.4 + 3.48i)T^{2} \)
17 \( 1 + (0.841 - 0.532i)T + (7.28 - 15.3i)T^{2} \)
19 \( 1 + (-5.07 - 2.93i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.70 + 8.15i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-5.04 + 6.18i)T + (-5.80 - 28.4i)T^{2} \)
31 \( 1 + (1.73 - 7.03i)T + (-27.4 - 14.4i)T^{2} \)
37 \( 1 + (-5.75 + 1.66i)T + (31.2 - 19.7i)T^{2} \)
41 \( 1 + (-0.718 + 8.90i)T + (-40.4 - 6.57i)T^{2} \)
43 \( 1 + (1.65 - 5.71i)T + (-36.3 - 22.9i)T^{2} \)
47 \( 1 + (2.96 - 3.35i)T + (-5.66 - 46.6i)T^{2} \)
53 \( 1 + (-0.831 - 0.436i)T + (30.1 + 43.6i)T^{2} \)
59 \( 1 + (4.64 - 10.9i)T + (-40.8 - 42.5i)T^{2} \)
61 \( 1 + (0.490 - 12.1i)T + (-60.8 - 4.90i)T^{2} \)
67 \( 1 + (2.87 + 0.586i)T + (61.6 + 26.2i)T^{2} \)
71 \( 1 + (-9.19 + 4.36i)T + (44.9 - 54.9i)T^{2} \)
73 \( 1 + (2.95 + 1.11i)T + (54.6 + 48.4i)T^{2} \)
79 \( 1 + (-6.64 - 5.88i)T + (9.52 + 78.4i)T^{2} \)
83 \( 1 + (-3.60 + 2.48i)T + (29.4 - 77.6i)T^{2} \)
89 \( 1 + (5.11 - 2.95i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.33 + 5.77i)T + (-26.9 - 93.1i)T^{2} \)
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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

−10.33100155033746077155066932783, −9.676535842693626636026616870167, −8.778785925673149155719640935160, −7.69609094592691290673409946921, −6.67794919919725280384929218481, −6.09618799858536900996404441644, −4.86415638087325712669271063975, −4.03478256598791225424144198316, −2.56972527651449328582007554858, −0.932897438279869359115376565953, 0.26086304887885256344625406450, 2.82823023830734743659557415296, 3.42568691581951804887043240143, 5.01031238586035272361667832484, 5.78416772794972692438791552982, 6.64254304838035392766763454894, 7.75044637514851745874477746601, 8.310449132861301820389510874901, 9.285516359689260898124017325763, 9.773705638555272465746274912678

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