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Properties

Label 2-845-5.4-c1-0-39
Degree $2$
Conductor $845$
Sign $0.644 - 0.764i$
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  + 1.18i·2-s − 0.345i·3-s + 0.590·4-s + (1.44 − 1.71i)5-s + 0.409·6-s + 2.02i·7-s + 3.07i·8-s + 2.88·9-s + (2.03 + 1.71i)10-s + 3.88·11-s − 0.203i·12-s − 2.40·14-s + (−0.590 − 0.497i)15-s − 2.47·16-s − 5.45i·17-s + 3.42i·18-s + ⋯
L(s)  = 1  + 0.839i·2-s − 0.199i·3-s + 0.295·4-s + (0.644 − 0.764i)5-s + 0.167·6-s + 0.767i·7-s + 1.08i·8-s + 0.960·9-s + (0.642 + 0.540i)10-s + 1.17·11-s − 0.0588i·12-s − 0.644·14-s + (−0.152 − 0.128i)15-s − 0.617·16-s − 1.32i·17-s + 0.806i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.644 - 0.764i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.644 - 0.764i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $0.644 - 0.764i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (339, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ 0.644 - 0.764i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.02596 + 0.942468i\)
\(L(\frac12)\) \(\approx\) \(2.02596 + 0.942468i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.44 + 1.71i)T \)
13 \( 1 \)
good2 \( 1 - 1.18iT - 2T^{2} \)
3 \( 1 + 0.345iT - 3T^{2} \)
7 \( 1 - 2.02iT - 7T^{2} \)
11 \( 1 - 3.88T + 11T^{2} \)
17 \( 1 + 5.45iT - 17T^{2} \)
19 \( 1 + 5.88T + 19T^{2} \)
23 \( 1 - 0.345iT - 23T^{2} \)
29 \( 1 + 3T + 29T^{2} \)
31 \( 1 - 1.18T + 31T^{2} \)
37 \( 1 - 5.45iT - 37T^{2} \)
41 \( 1 - 0.180T + 41T^{2} \)
43 \( 1 - 1.33iT - 43T^{2} \)
47 \( 1 - 12.2iT - 47T^{2} \)
53 \( 1 + 2.42iT - 53T^{2} \)
59 \( 1 - 7.06T + 59T^{2} \)
61 \( 1 - 6.76T + 61T^{2} \)
67 \( 1 + 4.40iT - 67T^{2} \)
71 \( 1 + 1.88T + 71T^{2} \)
73 \( 1 + 8.86iT - 73T^{2} \)
79 \( 1 + 11.1T + 79T^{2} \)
83 \( 1 + 7.83iT - 83T^{2} \)
89 \( 1 + 12.2T + 89T^{2} \)
97 \( 1 + 5.80iT - 97T^{2} \)
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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

−10.07310393715105649289204532932, −9.228052110636870824351900905803, −8.643781372431923272211433954354, −7.65045863119148302537792964593, −6.68608076033455997842954000819, −6.17468883623722454149471527034, −5.18212271102741509524849106030, −4.31415939638225285218739926140, −2.51277206659792964345025236724, −1.48496378270341589661839236468, 1.39143092142458939325694750191, 2.25321160537544245432158362737, 3.83418912887969198855496300148, 4.01886311347518242669195061253, 5.87233109217711076586493456930, 6.80707953338289271314501882211, 7.12178992421015175171972663511, 8.598426669013407386827849811812, 9.702136070077098388080860591493, 10.23931034259053501694066096058

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