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Properties

Label 2-845-65.8-c1-0-41
Degree $2$
Conductor $845$
Sign $0.372 + 0.928i$
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  − 2.64·2-s + (0.917 + 0.917i)3-s + 5.02·4-s + (1.30 − 1.81i)5-s + (−2.43 − 2.43i)6-s − 0.112i·7-s − 8.00·8-s − 1.31i·9-s + (−3.45 + 4.81i)10-s + (−1.31 + 1.31i)11-s + (4.60 + 4.60i)12-s + 0.297i·14-s + (2.86 − 0.470i)15-s + 11.1·16-s + (−1.93 − 1.93i)17-s + 3.49i·18-s + ⋯
L(s)  = 1  − 1.87·2-s + (0.529 + 0.529i)3-s + 2.51·4-s + (0.583 − 0.812i)5-s + (−0.992 − 0.992i)6-s − 0.0424i·7-s − 2.83·8-s − 0.439i·9-s + (−1.09 + 1.52i)10-s + (−0.395 + 0.395i)11-s + (1.32 + 1.32i)12-s + 0.0795i·14-s + (0.738 − 0.121i)15-s + 2.79·16-s + (−0.468 − 0.468i)17-s + 0.823i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.372 + 0.928i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.372 + 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $0.372 + 0.928i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (268, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ 0.372 + 0.928i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.627371 - 0.424212i\)
\(L(\frac12)\) \(\approx\) \(0.627371 - 0.424212i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.30 + 1.81i)T \)
13 \( 1 \)
good2 \( 1 + 2.64T + 2T^{2} \)
3 \( 1 + (-0.917 - 0.917i)T + 3iT^{2} \)
7 \( 1 + 0.112iT - 7T^{2} \)
11 \( 1 + (1.31 - 1.31i)T - 11iT^{2} \)
17 \( 1 + (1.93 + 1.93i)T + 17iT^{2} \)
19 \( 1 + (-4.92 + 4.92i)T - 19iT^{2} \)
23 \( 1 + (2.27 - 2.27i)T - 23iT^{2} \)
29 \( 1 + 4.65iT - 29T^{2} \)
31 \( 1 + (-0.624 - 0.624i)T + 31iT^{2} \)
37 \( 1 - 1.47iT - 37T^{2} \)
41 \( 1 + (3.83 + 3.83i)T + 41iT^{2} \)
43 \( 1 + (2.75 - 2.75i)T - 43iT^{2} \)
47 \( 1 + 0.345iT - 47T^{2} \)
53 \( 1 + (3.59 + 3.59i)T + 53iT^{2} \)
59 \( 1 + (-0.908 - 0.908i)T + 59iT^{2} \)
61 \( 1 + 2.78T + 61T^{2} \)
67 \( 1 - 0.144T + 67T^{2} \)
71 \( 1 + (-3.87 - 3.87i)T + 71iT^{2} \)
73 \( 1 + 9.06T + 73T^{2} \)
79 \( 1 + 15.1iT - 79T^{2} \)
83 \( 1 + 8.53iT - 83T^{2} \)
89 \( 1 + (0.402 + 0.402i)T + 89iT^{2} \)
97 \( 1 - 14.9T + 97T^{2} \)
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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

−9.833978917366256126175752234583, −9.160993399590112286334772512683, −8.729736862564699169401863058741, −7.78933339947488726454362124078, −6.95798672798740706824517474878, −5.92883312278947278663502003767, −4.68710673254408098349511474423, −3.08233995316064191633907333838, −2.01862890467020045879858504034, −0.64012863970570080140320745786, 1.47328268643258142764413857914, 2.34367164986690478067650330171, 3.25571070741672482053116192907, 5.55715521680673420923381599042, 6.48467457561728084081645644513, 7.27781173643710658875705960561, 7.938302122828119322827760376017, 8.584597100117341115151804586119, 9.477607230835999356842059015168, 10.30985095616773778689439271312

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