Properties

Label 2-637-13.12-c1-0-29
Degree $2$
Conductor $637$
Sign $0.645 - 0.763i$
Analytic cond. $5.08647$
Root an. cond. $2.25532$
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  + 1.52i·2-s + 2.99·3-s − 0.320·4-s − 2.94i·5-s + 4.56i·6-s + 2.55i·8-s + 5.97·9-s + 4.48·10-s + 2.37i·11-s − 0.960·12-s + (2.32 − 2.75i)13-s − 8.82i·15-s − 4.53·16-s − 5.36·17-s + 9.09i·18-s − 5.35i·19-s + ⋯
L(s)  = 1  + 1.07i·2-s + 1.72·3-s − 0.160·4-s − 1.31i·5-s + 1.86i·6-s + 0.904i·8-s + 1.99·9-s + 1.41·10-s + 0.714i·11-s − 0.277·12-s + (0.645 − 0.763i)13-s − 2.27i·15-s − 1.13·16-s − 1.30·17-s + 2.14i·18-s − 1.22i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.645 - 0.763i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.645 - 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.645 - 0.763i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (246, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.645 - 0.763i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.56281 + 1.18955i\)
\(L(\frac12)\) \(\approx\) \(2.56281 + 1.18955i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 + (-2.32 + 2.75i)T \)
good2 \( 1 - 1.52iT - 2T^{2} \)
3 \( 1 - 2.99T + 3T^{2} \)
5 \( 1 + 2.94iT - 5T^{2} \)
11 \( 1 - 2.37iT - 11T^{2} \)
17 \( 1 + 5.36T + 17T^{2} \)
19 \( 1 + 5.35iT - 19T^{2} \)
23 \( 1 + 2.79T + 23T^{2} \)
29 \( 1 - 0.585T + 29T^{2} \)
31 \( 1 - 8.33iT - 31T^{2} \)
37 \( 1 + 0.675iT - 37T^{2} \)
41 \( 1 - 11.2iT - 41T^{2} \)
43 \( 1 + 10.5T + 43T^{2} \)
47 \( 1 + 0.537iT - 47T^{2} \)
53 \( 1 - 7.90T + 53T^{2} \)
59 \( 1 - 6.14iT - 59T^{2} \)
61 \( 1 + 2.78T + 61T^{2} \)
67 \( 1 - 4.21iT - 67T^{2} \)
71 \( 1 - 1.25iT - 71T^{2} \)
73 \( 1 + 10.8iT - 73T^{2} \)
79 \( 1 - 6.29T + 79T^{2} \)
83 \( 1 + 1.74iT - 83T^{2} \)
89 \( 1 + 10.2iT - 89T^{2} \)
97 \( 1 + 9.90iT - 97T^{2} \)
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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

−10.37405199849286732898318984385, −9.249665052560998318501715481481, −8.662987897524039042228216622228, −8.271711087138003030355661456963, −7.35357183564073317787183389840, −6.51222807014816280652518381469, −5.06246457732926709922119877474, −4.38563725255992807933788996421, −2.90870078349666188480153496777, −1.71867153774401612596302557481, 1.90144432928700038317581042102, 2.52511065860498504359334667506, 3.61371559891088331184658480547, 3.93925938223440425758532672243, 6.29300190102832016000172169142, 7.00226947591868212302568220902, 8.021737864101883517680797290959, 8.883207633032254145643442209962, 9.720287575752902437308048980620, 10.46324702984974990391002009850

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