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Properties

Label 2-637-13.12-c1-0-36
Degree $2$
Conductor $637$
Sign $0.409 + 0.912i$
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  − 0.680i·2-s + 2.33·3-s + 1.53·4-s − 3.24i·5-s − 1.58i·6-s − 2.40i·8-s + 2.44·9-s − 2.20·10-s + 5.33i·11-s + 3.58·12-s + (1.47 + 3.28i)13-s − 7.57i·15-s + 1.43·16-s + 2.66·17-s − 1.66i·18-s − 0.696i·19-s + ⋯
L(s)  = 1  − 0.480i·2-s + 1.34·3-s + 0.768·4-s − 1.45i·5-s − 0.648i·6-s − 0.850i·8-s + 0.816·9-s − 0.698·10-s + 1.60i·11-s + 1.03·12-s + (0.409 + 0.912i)13-s − 1.95i·15-s + 0.359·16-s + 0.647·17-s − 0.392i·18-s − 0.159i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.409 + 0.912i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.409 + 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.409 + 0.912i$
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: \((2,\ 637,\ (\ :1/2),\ 0.409 + 0.912i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.29296 - 1.48374i\)
\(L(\frac12)\) \(\approx\) \(2.29296 - 1.48374i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.47 - 3.28i)T \)
good2 \( 1 + 0.680iT - 2T^{2} \)
3 \( 1 - 2.33T + 3T^{2} \)
5 \( 1 + 3.24iT - 5T^{2} \)
11 \( 1 - 5.33iT - 11T^{2} \)
17 \( 1 - 2.66T + 17T^{2} \)
19 \( 1 + 0.696iT - 19T^{2} \)
23 \( 1 + 8.96T + 23T^{2} \)
29 \( 1 + 5.28T + 29T^{2} \)
31 \( 1 + 5.45iT - 31T^{2} \)
37 \( 1 - 6.69iT - 37T^{2} \)
41 \( 1 + 2.21iT - 41T^{2} \)
43 \( 1 - 2.39T + 43T^{2} \)
47 \( 1 + 5.79iT - 47T^{2} \)
53 \( 1 + 4.71T + 53T^{2} \)
59 \( 1 - 2.98iT - 59T^{2} \)
61 \( 1 + 2.19T + 61T^{2} \)
67 \( 1 - 8.70iT - 67T^{2} \)
71 \( 1 - 8.56iT - 71T^{2} \)
73 \( 1 - 5.88iT - 73T^{2} \)
79 \( 1 - 0.910T + 79T^{2} \)
83 \( 1 + 11.7iT - 83T^{2} \)
89 \( 1 - 0.986iT - 89T^{2} \)
97 \( 1 - 15.3iT - 97T^{2} \)
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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

−9.978831426189988214703675608939, −9.661858703543773196336944680466, −8.757144382203248612024473500075, −7.905777783715725462375082357194, −7.19992780650646241225670462192, −5.87342450626245878806221715167, −4.44822061173224657381472819275, −3.74384598286330055350044875958, −2.22987508235861721673327092508, −1.61762516249177544862296570984, 2.09303030791743850628126878799, 3.21812649564746539988783648037, 3.44134354912218024155714599497, 5.79117687867777346961818204894, 6.22946850479194061690219044845, 7.52830848019789597993116743280, 7.88223020850484695192992101691, 8.693862556638016662889010913126, 9.919512740344064659516910737831, 10.79143455549795827233275411422

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