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Properties

Label 2-845-13.12-c1-0-1
Degree $2$
Conductor $845$
Sign $0.554 - 0.832i$
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  − 1.61i·2-s − 2.23·3-s − 0.618·4-s − i·5-s + 3.61i·6-s − 0.236i·7-s − 2.23i·8-s + 2.00·9-s − 1.61·10-s + 4.23i·11-s + 1.38·12-s − 0.381·14-s + 2.23i·15-s − 4.85·16-s − 5.47·17-s − 3.23i·18-s + ⋯
L(s)  = 1  − 1.14i·2-s − 1.29·3-s − 0.309·4-s − 0.447i·5-s + 1.47i·6-s − 0.0892i·7-s − 0.790i·8-s + 0.666·9-s − 0.511·10-s + 1.27i·11-s + 0.398·12-s − 0.102·14-s + 0.577i·15-s − 1.21·16-s − 1.32·17-s − 0.762i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.554 - 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.554 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $0.554 - 0.832i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (506, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ 0.554 - 0.832i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.180139 + 0.0964079i\)
\(L(\frac12)\) \(\approx\) \(0.180139 + 0.0964079i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + iT \)
13 \( 1 \)
good2 \( 1 + 1.61iT - 2T^{2} \)
3 \( 1 + 2.23T + 3T^{2} \)
7 \( 1 + 0.236iT - 7T^{2} \)
11 \( 1 - 4.23iT - 11T^{2} \)
17 \( 1 + 5.47T + 17T^{2} \)
19 \( 1 - 0.236iT - 19T^{2} \)
23 \( 1 + 8.23T + 23T^{2} \)
29 \( 1 - 1.47T + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 3iT - 37T^{2} \)
41 \( 1 - 5.94iT - 41T^{2} \)
43 \( 1 - 1.76T + 43T^{2} \)
47 \( 1 - 12.9iT - 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 - 12.7iT - 59T^{2} \)
61 \( 1 + 12.4T + 61T^{2} \)
67 \( 1 + 10.7iT - 67T^{2} \)
71 \( 1 - 1.76iT - 71T^{2} \)
73 \( 1 + 6iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 8.94iT - 83T^{2} \)
89 \( 1 + 9iT - 89T^{2} \)
97 \( 1 + 5.47iT - 97T^{2} \)
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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

−10.51864048623654723906586117497, −9.862005113454335200116331041251, −9.023958343932312895246664910869, −7.68372182912784287011691563106, −6.67101712324513810104952967780, −5.96978784545879862874709134939, −4.65141490598796595698260097667, −4.20891715640246204951854860809, −2.52864011569889320657716860193, −1.41501854811637285457450517566, 0.11575874798333552887712598928, 2.34344446055264310111109004415, 3.98889324682043975827640948007, 5.22530065618808066344853080596, 5.88978015003833419850346873108, 6.43162567682306514591990226315, 7.15847557643261841331104905357, 8.235870166479340505248521506005, 8.911339093207094463624898656450, 10.32938531009247012933180596636

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