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Properties

Label 2-126-63.25-c3-0-15
Degree $2$
Conductor $126$
Sign $0.971 + 0.236i$
Analytic cond. $7.43424$
Root an. cond. $2.72658$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + (−2.56 + 4.51i)3-s + 4·4-s + (8.32 − 14.4i)5-s + (−5.12 + 9.03i)6-s + (9.51 − 15.8i)7-s + 8·8-s + (−13.8 − 23.1i)9-s + (16.6 − 28.8i)10-s + (9.81 + 17.0i)11-s + (−10.2 + 18.0i)12-s + (18.2 + 31.6i)13-s + (19.0 − 31.7i)14-s + (43.7 + 74.5i)15-s + 16·16-s + (44.3 − 76.7i)17-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.493 + 0.869i)3-s + 0.5·4-s + (0.744 − 1.28i)5-s + (−0.348 + 0.615i)6-s + (0.513 − 0.858i)7-s + 0.353·8-s + (−0.512 − 0.858i)9-s + (0.526 − 0.911i)10-s + (0.269 + 0.466i)11-s + (−0.246 + 0.434i)12-s + (0.389 + 0.674i)13-s + (0.363 − 0.606i)14-s + (0.753 + 1.28i)15-s + 0.250·16-s + (0.632 − 1.09i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 + 0.236i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.971 + 0.236i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.971 + 0.236i$
Analytic conductor: \(7.43424\)
Root analytic conductor: \(2.72658\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :3/2),\ 0.971 + 0.236i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.46360 - 0.295279i\)
\(L(\frac12)\) \(\approx\) \(2.46360 - 0.295279i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2T \)
3 \( 1 + (2.56 - 4.51i)T \)
7 \( 1 + (-9.51 + 15.8i)T \)
good5 \( 1 + (-8.32 + 14.4i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-9.81 - 17.0i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-18.2 - 31.6i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (-44.3 + 76.7i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-37.5 - 65.0i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-19.4 + 33.7i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (122. - 212. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + 112.T + 2.97e4T^{2} \)
37 \( 1 + (206. + 357. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (39.5 + 68.4i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (232. - 402. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 - 402.T + 1.03e5T^{2} \)
53 \( 1 + (209. - 363. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 - 697.T + 2.05e5T^{2} \)
61 \( 1 + 232.T + 2.26e5T^{2} \)
67 \( 1 + 44.4T + 3.00e5T^{2} \)
71 \( 1 + 36.0T + 3.57e5T^{2} \)
73 \( 1 + (147. - 256. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + 1.25e3T + 4.93e5T^{2} \)
83 \( 1 + (228. - 396. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (-437. - 757. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (372. - 644. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−12.80798507875658274847192025721, −11.93154279105485624934600034235, −10.88651590786590434253555418142, −9.772565807695467797889977261763, −8.899093820462811757351435819045, −7.18556114653674520899154504864, −5.62652012481668004700044505763, −4.88811646960199079914228900234, −3.84513448700189684226545688134, −1.32967623221336667903295421458, 1.85893905367497105818392567975, 3.13961628195381157180351640350, 5.47796378064217572153783861302, 6.06224509292968326625964181520, 7.14464087607016022817259696239, 8.394232813023872164735437274239, 10.21334282684869101325854445699, 11.17453847450115806353515675651, 11.86862573325817758043717668958, 13.09791938122135168708690786436

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