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Properties

Label 2-845-65.37-c1-0-52
Degree $2$
Conductor $845$
Sign $0.851 + 0.523i$
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  + (−1.58 + 0.915i)2-s + (1.91 + 0.512i)3-s + (0.677 − 1.17i)4-s + (1.69 − 1.45i)5-s + (−3.50 + 0.939i)6-s + (1.76 − 3.06i)7-s − 1.18i·8-s + (0.803 + 0.463i)9-s + (−1.36 + 3.86i)10-s + (−3.74 − 1.00i)11-s + (1.89 − 1.89i)12-s + 6.48i·14-s + (3.99 − 1.91i)15-s + (2.43 + 4.22i)16-s + (−0.524 − 1.95i)17-s − 1.69·18-s + ⋯
L(s)  = 1  + (−1.12 + 0.647i)2-s + (1.10 + 0.296i)3-s + (0.338 − 0.586i)4-s + (0.759 − 0.650i)5-s + (−1.43 + 0.383i)6-s + (0.668 − 1.15i)7-s − 0.417i·8-s + (0.267 + 0.154i)9-s + (−0.430 + 1.22i)10-s + (−1.12 − 0.302i)11-s + (0.548 − 0.548i)12-s + 1.73i·14-s + (1.03 − 0.494i)15-s + (0.609 + 1.05i)16-s + (−0.127 − 0.474i)17-s − 0.400·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.851 + 0.523i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.851 + 0.523i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $0.851 + 0.523i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (427, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ 0.851 + 0.523i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.22895 - 0.347584i\)
\(L(\frac12)\) \(\approx\) \(1.22895 - 0.347584i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.69 + 1.45i)T \)
13 \( 1 \)
good2 \( 1 + (1.58 - 0.915i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (-1.91 - 0.512i)T + (2.59 + 1.5i)T^{2} \)
7 \( 1 + (-1.76 + 3.06i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (3.74 + 1.00i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (0.524 + 1.95i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-0.139 - 0.518i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (0.0788 - 0.294i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (-1.71 + 0.988i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.13 + 4.13i)T + 31iT^{2} \)
37 \( 1 + (-2.70 - 4.69i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.174 + 0.649i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-8.51 + 2.28i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + 9.75T + 47T^{2} \)
53 \( 1 + (-3.16 + 3.16i)T - 53iT^{2} \)
59 \( 1 + (-11.7 + 3.14i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-1.44 + 2.49i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.98 - 1.14i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (4.46 - 1.19i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 - 14.7iT - 73T^{2} \)
79 \( 1 - 1.59iT - 79T^{2} \)
83 \( 1 - 7.57T + 83T^{2} \)
89 \( 1 + (-1.21 + 4.54i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-15.4 - 8.91i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.880261727257276874000286191878, −9.145167622144313311158439701952, −8.348368281356304747805046868696, −7.923213358441146925355760423879, −7.13067014532073337970704141955, −5.90160403349277794072128335153, −4.74609962024195639579653752908, −3.67718330334952368975448560202, −2.26889023441531115661583985077, −0.78202258286069750546902579239, 1.81124811755786337625005788185, 2.34896420907734707760125727368, 3.08321452138816408287915723923, 5.08154423393561238569679262911, 5.87440373697325364717696175747, 7.35881034021701452257061666567, 8.015254528508575538502252166260, 8.802553062409282208904417909726, 9.227669281035982340793267457223, 10.19468967609923140447885734601

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