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Properties

Label 2-845-13.12-c1-0-30
Degree $2$
Conductor $845$
Sign $0.0304 - 0.999i$
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  + 1.53i·2-s + 2.88·3-s − 0.343·4-s + i·5-s + 4.42i·6-s − 3.86i·7-s + 2.53i·8-s + 5.33·9-s − 1.53·10-s + 4.75i·11-s − 0.992·12-s + 5.91·14-s + 2.88i·15-s − 4.56·16-s + 0.640·17-s + 8.16i·18-s + ⋯
L(s)  = 1  + 1.08i·2-s + 1.66·3-s − 0.171·4-s + 0.447i·5-s + 1.80i·6-s − 1.46i·7-s + 0.896i·8-s + 1.77·9-s − 0.484·10-s + 1.43i·11-s − 0.286·12-s + 1.58·14-s + 0.745i·15-s − 1.14·16-s + 0.155·17-s + 1.92i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0304 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0304 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $0.0304 - 0.999i$
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: \((2,\ 845,\ (\ :1/2),\ 0.0304 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.13421 + 2.07011i\)
\(L(\frac12)\) \(\approx\) \(2.13421 + 2.07011i\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.53iT - 2T^{2} \)
3 \( 1 - 2.88T + 3T^{2} \)
7 \( 1 + 3.86iT - 7T^{2} \)
11 \( 1 - 4.75iT - 11T^{2} \)
17 \( 1 - 0.640T + 17T^{2} \)
19 \( 1 + 4.70iT - 19T^{2} \)
23 \( 1 - 0.308T + 23T^{2} \)
29 \( 1 - 2.48T + 29T^{2} \)
31 \( 1 - 0.635iT - 31T^{2} \)
37 \( 1 - 5.15iT - 37T^{2} \)
41 \( 1 + 10.4iT - 41T^{2} \)
43 \( 1 + 8.56T + 43T^{2} \)
47 \( 1 - 2.89iT - 47T^{2} \)
53 \( 1 - 1.28T + 53T^{2} \)
59 \( 1 + 4.06iT - 59T^{2} \)
61 \( 1 - 0.335T + 61T^{2} \)
67 \( 1 + 0.721iT - 67T^{2} \)
71 \( 1 - 10.3iT - 71T^{2} \)
73 \( 1 + 13.1iT - 73T^{2} \)
79 \( 1 - 10.7T + 79T^{2} \)
83 \( 1 + 8.94iT - 83T^{2} \)
89 \( 1 - 0.0141iT - 89T^{2} \)
97 \( 1 + 10.1iT - 97T^{2} \)
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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

−10.13883725268881422293254880090, −9.394221088389401021314380841671, −8.414269822382158193303395711056, −7.67269084721298743956103789521, −7.08975987088591998712120319108, −6.70698234358234796332063314165, −4.93286313936498636329208696337, −4.09130008306269155163216199824, −2.95430851754214936995063996207, −1.87108933717430820141218749045, 1.45874850848498383932699729454, 2.49066137090932956375252244147, 3.15590150849121746908539268198, 3.96470629171878166019417308733, 5.46790359509862055326887140577, 6.53072240063991133732928174064, 8.012868770823601800187600534592, 8.430657236713925336976173789815, 9.228348258495156378520944512504, 9.755590621577531890586734452342

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