Properties

Label 2-637-91.74-c1-0-32
Degree $2$
Conductor $637$
Sign $-0.212 - 0.977i$
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  − 2.33·2-s + (−1.15 − 1.99i)3-s + 3.43·4-s + (−1.68 − 2.91i)5-s + (2.69 + 4.66i)6-s − 3.34·8-s + (−1.16 + 2.01i)9-s + (3.92 + 6.80i)10-s + (−1.16 − 2.01i)11-s + (−3.96 − 6.87i)12-s + (0.408 − 3.58i)13-s + (−3.89 + 6.74i)15-s + 0.933·16-s − 5.45·17-s + (2.71 − 4.70i)18-s + (3.58 − 6.20i)19-s + ⋯
L(s)  = 1  − 1.64·2-s + (−0.666 − 1.15i)3-s + 1.71·4-s + (−0.753 − 1.30i)5-s + (1.09 + 1.90i)6-s − 1.18·8-s + (−0.388 + 0.673i)9-s + (1.24 + 2.15i)10-s + (−0.351 − 0.608i)11-s + (−1.14 − 1.98i)12-s + (0.113 − 0.993i)13-s + (−1.00 + 1.74i)15-s + 0.233·16-s − 1.32·17-s + (0.640 − 1.10i)18-s + (0.822 − 1.42i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.212 - 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.212 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.212 - 0.977i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (165, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.212 - 0.977i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.160098 + 0.198752i\)
\(L(\frac12)\) \(\approx\) \(0.160098 + 0.198752i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.408 + 3.58i)T \)
good2 \( 1 + 2.33T + 2T^{2} \)
3 \( 1 + (1.15 + 1.99i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (1.68 + 2.91i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.16 + 2.01i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + 5.45T + 17T^{2} \)
19 \( 1 + (-3.58 + 6.20i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 6.45T + 23T^{2} \)
29 \( 1 + (-4.22 + 7.31i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.52 + 2.64i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 3.05T + 37T^{2} \)
41 \( 1 + (-0.468 + 0.812i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.04 - 3.54i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.73 - 2.99i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.17 + 2.02i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 7.24T + 59T^{2} \)
61 \( 1 + (3.19 - 5.53i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.30 + 3.99i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3.79 - 6.57i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1.03 + 1.79i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.79 - 6.57i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 2.89T + 83T^{2} \)
89 \( 1 - 13.1T + 89T^{2} \)
97 \( 1 + (-1.77 - 3.08i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.765734944433900610717918172726, −8.882589998448436677951226909338, −8.084852979540723616879157486073, −7.74080510275309435842453862535, −6.69051563505276894270792354193, −5.74956440649186877852129858700, −4.45991083730456013472506109378, −2.40748614510542649189598514245, −0.878688011360595874755261321742, −0.35894710697158135918754691277, 2.10360427796975208794920637830, 3.62595039159086142585410963860, 4.63224198209913954776989016824, 6.18806515062022866840856321310, 7.01630650603038303648272513617, 7.75187812483242678638045339334, 8.770538341942324467306015259132, 9.737639098360780839888041076290, 10.32385179607419292886159995388, 10.79374210418444086283556964912

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