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Properties

Label 2-637-49.39-c1-0-38
Degree $2$
Conductor $637$
Sign $-0.247 + 0.968i$
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  + (−1.75 − 0.264i)2-s + (2.20 − 1.50i)3-s + (1.09 + 0.337i)4-s + (0.122 + 1.63i)5-s + (−4.26 + 2.05i)6-s + (1.02 − 2.43i)7-s + (1.36 + 0.658i)8-s + (1.50 − 3.83i)9-s + (0.217 − 2.90i)10-s + (−1.32 − 3.38i)11-s + (2.91 − 0.900i)12-s + (−0.623 + 0.781i)13-s + (−2.44 + 4.00i)14-s + (2.73 + 3.42i)15-s + (−4.11 − 2.80i)16-s + (−5.25 − 4.87i)17-s + ⋯
L(s)  = 1  + (−1.23 − 0.186i)2-s + (1.27 − 0.867i)3-s + (0.546 + 0.168i)4-s + (0.0548 + 0.731i)5-s + (−1.74 + 0.838i)6-s + (0.387 − 0.921i)7-s + (0.483 + 0.232i)8-s + (0.501 − 1.27i)9-s + (0.0687 − 0.917i)10-s + (−0.400 − 1.02i)11-s + (0.842 − 0.259i)12-s + (−0.172 + 0.216i)13-s + (−0.652 + 1.07i)14-s + (0.704 + 0.883i)15-s + (−1.02 − 0.701i)16-s + (−1.27 − 1.18i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.247 + 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.247 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.247 + 0.968i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (235, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.247 + 0.968i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.690444 - 0.889166i\)
\(L(\frac12)\) \(\approx\) \(0.690444 - 0.889166i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-1.02 + 2.43i)T \)
13 \( 1 + (0.623 - 0.781i)T \)
good2 \( 1 + (1.75 + 0.264i)T + (1.91 + 0.589i)T^{2} \)
3 \( 1 + (-2.20 + 1.50i)T + (1.09 - 2.79i)T^{2} \)
5 \( 1 + (-0.122 - 1.63i)T + (-4.94 + 0.745i)T^{2} \)
11 \( 1 + (1.32 + 3.38i)T + (-8.06 + 7.48i)T^{2} \)
17 \( 1 + (5.25 + 4.87i)T + (1.27 + 16.9i)T^{2} \)
19 \( 1 + (-3.12 - 5.40i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.14 + 1.05i)T + (1.71 - 22.9i)T^{2} \)
29 \( 1 + (-1.85 + 8.13i)T + (-26.1 - 12.5i)T^{2} \)
31 \( 1 + (-1.13 + 1.96i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-8.43 + 2.60i)T + (30.5 - 20.8i)T^{2} \)
41 \( 1 + (-3.12 - 1.50i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (0.0979 - 0.0471i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (10.8 + 1.63i)T + (44.9 + 13.8i)T^{2} \)
53 \( 1 + (-1.40 - 0.431i)T + (43.7 + 29.8i)T^{2} \)
59 \( 1 + (0.566 - 7.56i)T + (-58.3 - 8.79i)T^{2} \)
61 \( 1 + (12.9 - 3.99i)T + (50.4 - 34.3i)T^{2} \)
67 \( 1 + (-2.89 + 5.01i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.131 - 0.576i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (-4.25 + 0.641i)T + (69.7 - 21.5i)T^{2} \)
79 \( 1 + (1.79 + 3.10i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (6.37 + 7.99i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (2.73 - 6.96i)T + (-65.2 - 60.5i)T^{2} \)
97 \( 1 - 18.3T + 97T^{2} \)
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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

−10.11699414242806585918655066044, −9.348639870113193251044764141011, −8.490746113156088171064522819313, −7.76756883237207919907612487623, −7.36556442069489958313581435807, −6.35110234145190815759417752191, −4.51446315309410705891662251875, −3.07952759735175064831218199660, −2.19019786922936556224780635091, −0.832083718959288565439059271858, 1.71885409966679041865371694128, 2.86656468439071680374805004471, 4.48447961970994531644404471154, 4.96281901023951545770450803529, 6.77155367859569403595732039646, 7.901693100895065625640256497996, 8.476114868137943349188288745086, 9.138021548787974964011434338629, 9.463159655329650851924492560160, 10.42525397581938601192681405051

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