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Properties

Label 2-312-104.77-c3-0-11
Degree $2$
Conductor $312$
Sign $0.0324 + 0.999i$
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  + (1.33 + 2.49i)2-s + 3i·3-s + (−4.41 + 6.67i)4-s − 12.3·5-s + (−7.47 + 4.01i)6-s + 26.6i·7-s + (−22.5 − 2.06i)8-s − 9·9-s + (−16.5 − 30.8i)10-s + 33.2·11-s + (−20.0 − 13.2i)12-s + (−46.5 + 5.78i)13-s + (−66.3 + 35.6i)14-s − 37.1i·15-s + (−25.0 − 58.8i)16-s + 16.3·17-s + ⋯
L(s)  = 1  + (0.473 + 0.880i)2-s + 0.577i·3-s + (−0.551 + 0.834i)4-s − 1.10·5-s + (−0.508 + 0.273i)6-s + 1.43i·7-s + (−0.995 − 0.0911i)8-s − 0.333·9-s + (−0.523 − 0.974i)10-s + 0.911·11-s + (−0.481 − 0.318i)12-s + (−0.992 + 0.123i)13-s + (−1.26 + 0.680i)14-s − 0.638i·15-s + (−0.391 − 0.920i)16-s + 0.232·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0324 + 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0324 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(312\)    =    \(2^{3} \cdot 3 \cdot 13\)
Sign: $0.0324 + 0.999i$
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: \((2,\ 312,\ (\ :3/2),\ 0.0324 + 0.999i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.6891599119\)
\(L(\frac12)\) \(\approx\) \(0.6891599119\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.33 - 2.49i)T \)
3 \( 1 - 3iT \)
13 \( 1 + (46.5 - 5.78i)T \)
good5 \( 1 + 12.3T + 125T^{2} \)
7 \( 1 - 26.6iT - 343T^{2} \)
11 \( 1 - 33.2T + 1.33e3T^{2} \)
17 \( 1 - 16.3T + 4.91e3T^{2} \)
19 \( 1 - 43.9T + 6.85e3T^{2} \)
23 \( 1 - 60.7T + 1.21e4T^{2} \)
29 \( 1 + 218. iT - 2.43e4T^{2} \)
31 \( 1 - 12.9iT - 2.97e4T^{2} \)
37 \( 1 + 295.T + 5.06e4T^{2} \)
41 \( 1 - 241. iT - 6.89e4T^{2} \)
43 \( 1 + 290. iT - 7.95e4T^{2} \)
47 \( 1 - 385. iT - 1.03e5T^{2} \)
53 \( 1 - 220. iT - 1.48e5T^{2} \)
59 \( 1 + 310.T + 2.05e5T^{2} \)
61 \( 1 + 156. iT - 2.26e5T^{2} \)
67 \( 1 - 586.T + 3.00e5T^{2} \)
71 \( 1 - 1.14e3iT - 3.57e5T^{2} \)
73 \( 1 + 792. iT - 3.89e5T^{2} \)
79 \( 1 + 1.06e3T + 4.93e5T^{2} \)
83 \( 1 + 1.48e3T + 5.71e5T^{2} \)
89 \( 1 - 216. iT - 7.04e5T^{2} \)
97 \( 1 - 789. iT - 9.12e5T^{2} \)
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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

−11.92328611804066835097325241459, −11.51700674808193993123913002935, −9.715523044799194585388695253270, −8.964306702231399949270750136261, −8.122631613894856443413282903959, −7.13353500319341920379108773855, −5.93343731261589068181145456067, −4.96320845965010974710007167497, −3.98368857456383113234570758515, −2.83786875703432138912879938984, 0.22992649872063810860536264111, 1.38493062807427659442948059816, 3.24263538582534763004272469020, 4.05736577885475353062143549469, 5.14914984704579331673185596259, 6.82813291635321989146808063840, 7.44078326485470813252915145961, 8.692346309048661151341420551120, 9.874702260463115788718371121181, 10.78811692370588538774984214877

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