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Properties

Label 2-845-169.116-c1-0-12
Degree $2$
Conductor $845$
Sign $0.971 - 0.237i$
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  + (−0.430 − 0.485i)2-s + (−0.374 − 0.986i)3-s + (0.190 − 1.56i)4-s + (0.239 + 0.970i)5-s + (−0.318 + 0.605i)6-s + (−2.62 + 1.81i)7-s + (−1.91 + 1.31i)8-s + (1.41 − 1.25i)9-s + (0.368 − 0.533i)10-s + (−2.76 + 3.12i)11-s + (−1.61 + 0.398i)12-s + (3.11 + 1.81i)13-s + (2.01 + 0.495i)14-s + (0.868 − 0.599i)15-s + (−1.60 − 0.395i)16-s + (3.19 + 4.63i)17-s + ⋯
L(s)  = 1  + (−0.304 − 0.343i)2-s + (−0.216 − 0.569i)3-s + (0.0951 − 0.783i)4-s + (0.107 + 0.434i)5-s + (−0.129 + 0.247i)6-s + (−0.993 + 0.685i)7-s + (−0.675 + 0.466i)8-s + (0.470 − 0.417i)9-s + (0.116 − 0.168i)10-s + (−0.833 + 0.941i)11-s + (−0.467 + 0.115i)12-s + (0.864 + 0.502i)13-s + (0.537 + 0.132i)14-s + (0.224 − 0.154i)15-s + (−0.401 − 0.0988i)16-s + (0.775 + 1.12i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 - 0.237i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.971 - 0.237i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $0.971 - 0.237i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (116, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ 0.971 - 0.237i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.955351 + 0.115015i\)
\(L(\frac12)\) \(\approx\) \(0.955351 + 0.115015i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.239 - 0.970i)T \)
13 \( 1 + (-3.11 - 1.81i)T \)
good2 \( 1 + (0.430 + 0.485i)T + (-0.241 + 1.98i)T^{2} \)
3 \( 1 + (0.374 + 0.986i)T + (-2.24 + 1.98i)T^{2} \)
7 \( 1 + (2.62 - 1.81i)T + (2.48 - 6.54i)T^{2} \)
11 \( 1 + (2.76 - 3.12i)T + (-1.32 - 10.9i)T^{2} \)
17 \( 1 + (-3.19 - 4.63i)T + (-6.02 + 15.8i)T^{2} \)
19 \( 1 - 1.49iT - 19T^{2} \)
23 \( 1 - 5.81T + 23T^{2} \)
29 \( 1 + (-1.50 + 1.32i)T + (3.49 - 28.7i)T^{2} \)
31 \( 1 + (0.452 - 0.861i)T + (-17.6 - 25.5i)T^{2} \)
37 \( 1 + (1.23 - 2.35i)T + (-21.0 - 30.4i)T^{2} \)
41 \( 1 + (-5.97 + 2.26i)T + (30.6 - 27.1i)T^{2} \)
43 \( 1 + (5.18 - 2.72i)T + (24.4 - 35.3i)T^{2} \)
47 \( 1 + (5.52 - 0.670i)T + (45.6 - 11.2i)T^{2} \)
53 \( 1 + (-3.99 - 5.78i)T + (-18.7 + 49.5i)T^{2} \)
59 \( 1 + (1.99 + 8.08i)T + (-52.2 + 27.4i)T^{2} \)
61 \( 1 + (-0.123 + 0.179i)T + (-21.6 - 57.0i)T^{2} \)
67 \( 1 + (-11.5 + 1.39i)T + (65.0 - 16.0i)T^{2} \)
71 \( 1 + (4.90 - 1.86i)T + (53.1 - 47.0i)T^{2} \)
73 \( 1 + (4.55 - 5.14i)T + (-8.79 - 72.4i)T^{2} \)
79 \( 1 + (0.601 + 4.95i)T + (-76.7 + 18.9i)T^{2} \)
83 \( 1 + (-14.2 - 5.40i)T + (62.1 + 55.0i)T^{2} \)
89 \( 1 - 14.5iT - 89T^{2} \)
97 \( 1 + (0.460 - 1.86i)T + (-85.8 - 45.0i)T^{2} \)
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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

−10.12071514546307177472252590891, −9.631338628545118953310884695945, −8.749650193279943235638411741358, −7.56457326605183355351171919591, −6.47513092211170112999267794738, −6.20248772943191561298019723583, −5.11608740360752378461017834719, −3.57615107889684085952449082279, −2.38316741803412965322743427453, −1.29899578768562512090690437163, 0.58739257583101941328780880650, 2.98024784633758826054967611604, 3.60944636916104503357434745794, 4.83983082900044456119348803785, 5.77660497110469221410540069076, 6.90085581938562404523233330253, 7.59709651666075152184642058291, 8.480814389307562517540171564534, 9.296210489493210100039593165387, 10.09309308983333133120708725999

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