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Properties

Label 2-637-13.4-c1-0-31
Degree $2$
Conductor $637$
Sign $0.203 + 0.979i$
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  + (1.19 + 0.689i)2-s + (−1.44 + 2.49i)3-s + (−0.0491 − 0.0850i)4-s − 0.805i·5-s + (−3.44 + 1.98i)6-s − 2.89i·8-s + (−2.65 − 4.59i)9-s + (0.555 − 0.962i)10-s + (−4.56 − 2.63i)11-s + 0.282·12-s + (−2.36 − 2.72i)13-s + (2.01 + 1.16i)15-s + (1.89 − 3.28i)16-s + (−0.280 − 0.485i)17-s − 7.31i·18-s + (−5.06 + 2.92i)19-s + ⋯
L(s)  = 1  + (0.844 + 0.487i)2-s + (−0.831 + 1.44i)3-s + (−0.0245 − 0.0425i)4-s − 0.360i·5-s + (−1.40 + 0.811i)6-s − 1.02i·8-s + (−0.883 − 1.53i)9-s + (0.175 − 0.304i)10-s + (−1.37 − 0.794i)11-s + 0.0816·12-s + (−0.656 − 0.754i)13-s + (0.519 + 0.299i)15-s + (0.474 − 0.821i)16-s + (−0.0679 − 0.117i)17-s − 1.72i·18-s + (−1.16 + 0.670i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.203 + 0.979i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.203 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.203 + 0.979i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (589, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.203 + 0.979i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.401458 - 0.326593i\)
\(L(\frac12)\) \(\approx\) \(0.401458 - 0.326593i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.36 + 2.72i)T \)
good2 \( 1 + (-1.19 - 0.689i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (1.44 - 2.49i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + 0.805iT - 5T^{2} \)
11 \( 1 + (4.56 + 2.63i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (0.280 + 0.485i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (5.06 - 2.92i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.802 - 1.38i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.14 - 1.97i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 3.47iT - 31T^{2} \)
37 \( 1 + (1.07 + 0.620i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.803 - 0.463i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.22 - 3.85i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 3.84iT - 47T^{2} \)
53 \( 1 - 5.45T + 53T^{2} \)
59 \( 1 + (9.52 - 5.49i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.65 + 6.32i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.36 + 3.67i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (8.06 - 4.65i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 5.00iT - 73T^{2} \)
79 \( 1 - 11.3T + 79T^{2} \)
83 \( 1 - 5.81iT - 83T^{2} \)
89 \( 1 + (4.33 + 2.50i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-9.22 + 5.32i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.48544239841915542505527469583, −9.789571868309200010104009827564, −8.805019421595515386141413016129, −7.66037199848629603168636677289, −6.22162751905661424098819787934, −5.54979978607100395069444874697, −4.98385660266873199069230775669, −4.20722539697737036563270188056, −3.10138150253513707521576275583, −0.22121772897539359828661294894, 2.02496050926724903032143915242, 2.69574191918674271169498993247, 4.46892588985080529461560444792, 5.18572490176657279708372329604, 6.27093642654323376304810662342, 7.14708119854266204176086073328, 7.79744137923727035144993628523, 8.859059249381855460117968968873, 10.45250298624503840922337207233, 10.99287176881126427525843840735

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