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Properties

Label 2-845-13.12-c1-0-6
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  − 0.445i·2-s − 3.24·3-s + 1.80·4-s + i·5-s + 1.44i·6-s + 3.24i·7-s − 1.69i·8-s + 7.54·9-s + 0.445·10-s − 0.692i·11-s − 5.85·12-s + 1.44·14-s − 3.24i·15-s + 2.85·16-s − 3.74·17-s − 3.35i·18-s + ⋯
L(s)  = 1  − 0.314i·2-s − 1.87·3-s + 0.900·4-s + 0.447i·5-s + 0.589i·6-s + 1.22i·7-s − 0.598i·8-s + 2.51·9-s + 0.140·10-s − 0.208i·11-s − 1.68·12-s + 0.386·14-s − 0.838i·15-s + 0.712·16-s − 0.907·17-s − 0.791i·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

\(L(1)\) \(\approx\) \(0.540231 + 0.556959i\)
\(L(\frac12)\) \(\approx\) \(0.540231 + 0.556959i\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.445iT - 2T^{2} \)
3 \( 1 + 3.24T + 3T^{2} \)
7 \( 1 - 3.24iT - 7T^{2} \)
11 \( 1 + 0.692iT - 11T^{2} \)
17 \( 1 + 3.74T + 17T^{2} \)
19 \( 1 - 1.53iT - 19T^{2} \)
23 \( 1 - 1.22T + 23T^{2} \)
29 \( 1 + 6.07T + 29T^{2} \)
31 \( 1 - 8.45iT - 31T^{2} \)
37 \( 1 + 1.89iT - 37T^{2} \)
41 \( 1 - 0.457iT - 41T^{2} \)
43 \( 1 - 6.19T + 43T^{2} \)
47 \( 1 - 11.5iT - 47T^{2} \)
53 \( 1 - 0.801T + 53T^{2} \)
59 \( 1 - 6.60iT - 59T^{2} \)
61 \( 1 + 4.19T + 61T^{2} \)
67 \( 1 - 13.8iT - 67T^{2} \)
71 \( 1 - 9.87iT - 71T^{2} \)
73 \( 1 - 8.05iT - 73T^{2} \)
79 \( 1 + 16.5T + 79T^{2} \)
83 \( 1 + 6.17iT - 83T^{2} \)
89 \( 1 + 10.5iT - 89T^{2} \)
97 \( 1 - 3.45iT - 97T^{2} \)
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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

−10.73144840268496850956369976885, −9.978765167449018644990384297310, −8.907989567064191841834469792832, −7.43125214208712126414458005893, −6.75798617376700624485115770543, −5.93094639451320885289423707225, −5.51253813142228832628417221426, −4.21812036314208985947013128664, −2.73134804872679795630570967802, −1.48101973210686615876398255151, 0.48368720322470370609101972656, 1.83169866127678943973705320566, 3.95107095614143619059865110303, 4.81527995043297624951787616232, 5.70551436352173437536847099579, 6.51586499480523265007769156623, 7.14057183024163153118166074574, 7.80892311801244108397924486386, 9.398576972309477289160170513089, 10.34819384991572122882262048233

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