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Properties

Label 2-312-104.77-c3-0-26
Degree $2$
Conductor $312$
Sign $0.292 - 0.956i$
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  + (0.0631 + 2.82i)2-s − 3i·3-s + (−7.99 + 0.357i)4-s − 11.9·5-s + (8.48 − 0.189i)6-s − 11.1i·7-s + (−1.51 − 22.5i)8-s − 9·9-s + (−0.756 − 33.8i)10-s − 24.4·11-s + (1.07 + 23.9i)12-s + (−16.6 + 43.8i)13-s + (31.6 − 0.706i)14-s + 35.9i·15-s + (63.7 − 5.70i)16-s + 68.2·17-s + ⋯
L(s)  = 1  + (0.0223 + 0.999i)2-s − 0.577i·3-s + (−0.999 + 0.0446i)4-s − 1.07·5-s + (0.577 − 0.0128i)6-s − 0.604i·7-s + (−0.0669 − 0.997i)8-s − 0.333·9-s + (−0.0239 − 1.07i)10-s − 0.669·11-s + (0.0257 + 0.576i)12-s + (−0.355 + 0.934i)13-s + (0.604 − 0.0134i)14-s + 0.618i·15-s + (0.996 − 0.0891i)16-s + 0.974·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.292 - 0.956i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.292 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(312\)    =    \(2^{3} \cdot 3 \cdot 13\)
Sign: $0.292 - 0.956i$
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: \((2,\ 312,\ (\ :3/2),\ 0.292 - 0.956i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.056267220\)
\(L(\frac12)\) \(\approx\) \(1.056267220\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0631 - 2.82i)T \)
3 \( 1 + 3iT \)
13 \( 1 + (16.6 - 43.8i)T \)
good5 \( 1 + 11.9T + 125T^{2} \)
7 \( 1 + 11.1iT - 343T^{2} \)
11 \( 1 + 24.4T + 1.33e3T^{2} \)
17 \( 1 - 68.2T + 4.91e3T^{2} \)
19 \( 1 - 29.3T + 6.85e3T^{2} \)
23 \( 1 - 197.T + 1.21e4T^{2} \)
29 \( 1 - 184. iT - 2.43e4T^{2} \)
31 \( 1 + 32.3iT - 2.97e4T^{2} \)
37 \( 1 - 81.8T + 5.06e4T^{2} \)
41 \( 1 - 159. iT - 6.89e4T^{2} \)
43 \( 1 + 400. iT - 7.95e4T^{2} \)
47 \( 1 - 183. iT - 1.03e5T^{2} \)
53 \( 1 - 220. iT - 1.48e5T^{2} \)
59 \( 1 - 464.T + 2.05e5T^{2} \)
61 \( 1 - 182. iT - 2.26e5T^{2} \)
67 \( 1 + 292.T + 3.00e5T^{2} \)
71 \( 1 - 913. iT - 3.57e5T^{2} \)
73 \( 1 + 300. iT - 3.89e5T^{2} \)
79 \( 1 + 61.6T + 4.93e5T^{2} \)
83 \( 1 - 381.T + 5.71e5T^{2} \)
89 \( 1 + 172. iT - 7.04e5T^{2} \)
97 \( 1 + 933. iT - 9.12e5T^{2} \)
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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

−11.59360644686961799736764004594, −10.47044247302731738024323367772, −9.238760211121158247772420434840, −8.269532713149982705653952532168, −7.31751729998919280187321467982, −7.05163277700182162672599948190, −5.53052006007092475726718159284, −4.47260867472108062548856580739, −3.28236511013811096299323105289, −0.873581271658875457501815729818, 0.57501740409025770203239732198, 2.71245713261597840956784609074, 3.53906435177520969497534979974, 4.80647776535702995879217479533, 5.58973340132067312749575438354, 7.62439704519346621585300980417, 8.321951507165944838208766995279, 9.392934503917987432752847616051, 10.25138423640471604559609172088, 11.11525704040506633920349460638

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