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Properties

Label 2-312-104.77-c3-0-54
Degree $2$
Conductor $312$
Sign $0.839 + 0.542i$
Analytic cond. $18.4085$
Root an. cond. $4.29052$
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  + (−1.61 + 2.32i)2-s + 3i·3-s + (−2.78 − 7.50i)4-s + 1.96·5-s + (−6.96 − 4.84i)6-s + 7.12i·7-s + (21.9 + 5.66i)8-s − 9·9-s + (−3.16 + 4.55i)10-s − 39.6·11-s + (22.5 − 8.34i)12-s + (14.7 − 44.4i)13-s + (−16.5 − 11.5i)14-s + 5.88i·15-s + (−48.5 + 41.7i)16-s − 20.3·17-s + ⋯
L(s)  = 1  + (−0.571 + 0.820i)2-s + 0.577i·3-s + (−0.347 − 0.937i)4-s + 0.175·5-s + (−0.473 − 0.329i)6-s + 0.384i·7-s + (0.968 + 0.250i)8-s − 0.333·9-s + (−0.100 + 0.144i)10-s − 1.08·11-s + (0.541 − 0.200i)12-s + (0.315 − 0.948i)13-s + (−0.315 − 0.219i)14-s + 0.101i·15-s + (−0.758 + 0.651i)16-s − 0.291·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.839 + 0.542i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.839 + 0.542i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(312\)    =    \(2^{3} \cdot 3 \cdot 13\)
Sign: $0.839 + 0.542i$
Analytic conductor: \(18.4085\)
Root analytic conductor: \(4.29052\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{312} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 312,\ (\ :3/2),\ 0.839 + 0.542i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.7133178946\)
\(L(\frac12)\) \(\approx\) \(0.7133178946\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.61 - 2.32i)T \)
3 \( 1 - 3iT \)
13 \( 1 + (-14.7 + 44.4i)T \)
good5 \( 1 - 1.96T + 125T^{2} \)
7 \( 1 - 7.12iT - 343T^{2} \)
11 \( 1 + 39.6T + 1.33e3T^{2} \)
17 \( 1 + 20.3T + 4.91e3T^{2} \)
19 \( 1 - 78.7T + 6.85e3T^{2} \)
23 \( 1 + 108.T + 1.21e4T^{2} \)
29 \( 1 + 306. iT - 2.43e4T^{2} \)
31 \( 1 + 122. iT - 2.97e4T^{2} \)
37 \( 1 - 238.T + 5.06e4T^{2} \)
41 \( 1 + 113. iT - 6.89e4T^{2} \)
43 \( 1 + 443. iT - 7.95e4T^{2} \)
47 \( 1 - 435. iT - 1.03e5T^{2} \)
53 \( 1 - 496. iT - 1.48e5T^{2} \)
59 \( 1 - 868.T + 2.05e5T^{2} \)
61 \( 1 + 355. iT - 2.26e5T^{2} \)
67 \( 1 + 792.T + 3.00e5T^{2} \)
71 \( 1 - 208. iT - 3.57e5T^{2} \)
73 \( 1 + 1.07e3iT - 3.89e5T^{2} \)
79 \( 1 - 1.22e3T + 4.93e5T^{2} \)
83 \( 1 + 701.T + 5.71e5T^{2} \)
89 \( 1 + 14.2iT - 7.04e5T^{2} \)
97 \( 1 + 1.16e3iT - 9.12e5T^{2} \)
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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

−10.79941850948790617531010424126, −10.02212181201847391274902334783, −9.348613688996158780676919053423, −8.140416280802092304621334641135, −7.63464408454512468432426796427, −5.94995907545602359554610734230, −5.54360285678784686736392194274, −4.19597074195738092520929621361, −2.41965390304137052393937754127, −0.33590818090604311821022457188, 1.28581123225543430000125773896, 2.49965439163537564670914243944, 3.81935324596234666655728866274, 5.22171390866517861031667428914, 6.75670391082269502393012231951, 7.66796210710607373256317197129, 8.519536800097635106584575841743, 9.573439856570184047806313444596, 10.42084915929280399317000585529, 11.35065486631765691353245851847

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