Properties

Label 2-637-13.12-c1-0-28
Degree $2$
Conductor $637$
Sign $-0.554 + 0.832i$
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  − 2i·2-s + 2·3-s − 2·4-s + i·5-s − 4i·6-s + 9-s + 2·10-s − 2i·11-s − 4·12-s + (2 − 3i)13-s + 2i·15-s − 4·16-s + 6·17-s − 2i·18-s − 3i·19-s − 2i·20-s + ⋯
L(s)  = 1  − 1.41i·2-s + 1.15·3-s − 4-s + 0.447i·5-s − 1.63i·6-s + 0.333·9-s + 0.632·10-s − 0.603i·11-s − 1.15·12-s + (0.554 − 0.832i)13-s + 0.516i·15-s − 16-s + 1.45·17-s − 0.471i·18-s − 0.688i·19-s − 0.447i·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.554 + 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.554 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.554 + 0.832i$
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: \((2,\ 637,\ (\ :1/2),\ -0.554 + 0.832i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.01313 - 1.89306i\)
\(L(\frac12)\) \(\approx\) \(1.01313 - 1.89306i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2 + 3i)T \)
good2 \( 1 + 2iT - 2T^{2} \)
3 \( 1 - 2T + 3T^{2} \)
5 \( 1 - iT - 5T^{2} \)
11 \( 1 + 2iT - 11T^{2} \)
17 \( 1 - 6T + 17T^{2} \)
19 \( 1 + 3iT - 19T^{2} \)
23 \( 1 + 3T + 23T^{2} \)
29 \( 1 - 3T + 29T^{2} \)
31 \( 1 + 3iT - 31T^{2} \)
37 \( 1 - 6iT - 37T^{2} \)
41 \( 1 - 10iT - 41T^{2} \)
43 \( 1 - T + 43T^{2} \)
47 \( 1 + 11iT - 47T^{2} \)
53 \( 1 + 9T + 53T^{2} \)
59 \( 1 - 8iT - 59T^{2} \)
61 \( 1 - 8T + 61T^{2} \)
67 \( 1 - 12iT - 67T^{2} \)
71 \( 1 - 14iT - 71T^{2} \)
73 \( 1 - 9iT - 73T^{2} \)
79 \( 1 + 9T + 79T^{2} \)
83 \( 1 - 11iT - 83T^{2} \)
89 \( 1 - 5iT - 89T^{2} \)
97 \( 1 + 9iT - 97T^{2} \)
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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

−10.18803269324558568594717069969, −9.746101007605975765124687230574, −8.606407045003369079509021139899, −8.088788842302387893616119710109, −6.85266458704210460933678725820, −5.56138909409872972110325621622, −4.03358376185266144617444207717, −3.08957182611120525165091376776, −2.73064689323856064386024325161, −1.15690658711596644277879688543, 1.88813653210403727742605182787, 3.40767769512006132825402228318, 4.54523800628492560613720280741, 5.60119860321301535171771274622, 6.52707027617185373735316645633, 7.61738763057747751025397236972, 8.034339538881456639547981720508, 8.950676614316671573941948696336, 9.430037758233488900961057905407, 10.64234353303246160100060382416

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