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Properties

Label 2-1617-77.76-c1-0-21
Degree $2$
Conductor $1617$
Sign $0.222 - 0.974i$
Analytic cond. $12.9118$
Root an. cond. $3.59330$
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  + 0.315i·2-s − i·3-s + 1.90·4-s + 1.47i·5-s + 0.315·6-s + 1.22i·8-s − 9-s − 0.463·10-s + (−1.23 + 3.07i)11-s − 1.90i·12-s − 6.26·13-s + 1.47·15-s + 3.41·16-s + 5.48·17-s − 0.315i·18-s − 1.26·19-s + ⋯
L(s)  = 1  + 0.222i·2-s − 0.577i·3-s + 0.950·4-s + 0.658i·5-s + 0.128·6-s + 0.434i·8-s − 0.333·9-s − 0.146·10-s + (−0.372 + 0.927i)11-s − 0.548i·12-s − 1.73·13-s + 0.380·15-s + 0.853·16-s + 1.33·17-s − 0.0742i·18-s − 0.289·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.222 - 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.222 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1617\)    =    \(3 \cdot 7^{2} \cdot 11\)
Sign: $0.222 - 0.974i$
Analytic conductor: \(12.9118\)
Root analytic conductor: \(3.59330\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1617} (538, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1617,\ (\ :1/2),\ 0.222 - 0.974i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.809236963\)
\(L(\frac12)\) \(\approx\) \(1.809236963\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + iT \)
7 \( 1 \)
11 \( 1 + (1.23 - 3.07i)T \)
good2 \( 1 - 0.315iT - 2T^{2} \)
5 \( 1 - 1.47iT - 5T^{2} \)
13 \( 1 + 6.26T + 13T^{2} \)
17 \( 1 - 5.48T + 17T^{2} \)
19 \( 1 + 1.26T + 19T^{2} \)
23 \( 1 - 1.95T + 23T^{2} \)
29 \( 1 - 7.46iT - 29T^{2} \)
31 \( 1 - 5.39iT - 31T^{2} \)
37 \( 1 + 2.83T + 37T^{2} \)
41 \( 1 + 4.11T + 41T^{2} \)
43 \( 1 - 7.49iT - 43T^{2} \)
47 \( 1 - 2.91iT - 47T^{2} \)
53 \( 1 - 11.1T + 53T^{2} \)
59 \( 1 + 2.70iT - 59T^{2} \)
61 \( 1 - 9.39T + 61T^{2} \)
67 \( 1 + 11.0T + 67T^{2} \)
71 \( 1 - 13.1T + 71T^{2} \)
73 \( 1 + 16.7T + 73T^{2} \)
79 \( 1 + 1.39iT - 79T^{2} \)
83 \( 1 + 0.851T + 83T^{2} \)
89 \( 1 + 5.85iT - 89T^{2} \)
97 \( 1 - 12.4iT - 97T^{2} \)
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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

−9.790648598947694922188959727876, −8.550076201875304078195217750497, −7.58162244669424753943690575925, −7.14537748218100728776510680423, −6.72341931179854874397828311290, −5.53211348188322164600428409166, −4.87682925987173044263249893456, −3.18214090302887523976485704580, −2.58991960875557820781650065570, −1.53401680819182807700906973374, 0.66387624571851898923681745474, 2.24678058131966224380491332428, 3.06902301060689267928990744013, 4.10746890511384589941667004741, 5.25967449726257784989409018218, 5.70024443610435398448892340004, 6.89351565502417790814392865883, 7.68865656844120561549317065773, 8.411469010256618908503824021373, 9.387037728927417577861333172039

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