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Properties

Label 2-312-104.77-c3-0-75
Degree $2$
Conductor $312$
Sign $-0.765 + 0.644i$
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  + (2.30 − 1.63i)2-s + 3i·3-s + (2.62 − 7.55i)4-s + 3.15·5-s + (4.91 + 6.91i)6-s − 31.8i·7-s + (−6.33 − 21.7i)8-s − 9·9-s + (7.28 − 5.17i)10-s − 30.3·11-s + (22.6 + 7.87i)12-s + (−42.8 + 18.9i)13-s + (−52.1 − 73.3i)14-s + 9.47i·15-s + (−50.2 − 39.6i)16-s + 33.4·17-s + ⋯
L(s)  = 1  + (0.814 − 0.579i)2-s + 0.577i·3-s + (0.328 − 0.944i)4-s + 0.282·5-s + (0.334 + 0.470i)6-s − 1.71i·7-s + (−0.280 − 0.959i)8-s − 0.333·9-s + (0.230 − 0.163i)10-s − 0.831·11-s + (0.545 + 0.189i)12-s + (−0.914 + 0.403i)13-s + (−0.995 − 1.40i)14-s + 0.163i·15-s + (−0.784 − 0.620i)16-s + 0.477·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.765 + 0.644i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.765 + 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(312\)    =    \(2^{3} \cdot 3 \cdot 13\)
Sign: $-0.765 + 0.644i$
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: \((2,\ 312,\ (\ :3/2),\ -0.765 + 0.644i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.181363892\)
\(L(\frac12)\) \(\approx\) \(2.181363892\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.30 + 1.63i)T \)
3 \( 1 - 3iT \)
13 \( 1 + (42.8 - 18.9i)T \)
good5 \( 1 - 3.15T + 125T^{2} \)
7 \( 1 + 31.8iT - 343T^{2} \)
11 \( 1 + 30.3T + 1.33e3T^{2} \)
17 \( 1 - 33.4T + 4.91e3T^{2} \)
19 \( 1 + 32.4T + 6.85e3T^{2} \)
23 \( 1 - 92.1T + 1.21e4T^{2} \)
29 \( 1 - 11.6iT - 2.43e4T^{2} \)
31 \( 1 + 328. iT - 2.97e4T^{2} \)
37 \( 1 - 271.T + 5.06e4T^{2} \)
41 \( 1 + 153. iT - 6.89e4T^{2} \)
43 \( 1 - 26.1iT - 7.95e4T^{2} \)
47 \( 1 - 72.2iT - 1.03e5T^{2} \)
53 \( 1 + 666. iT - 1.48e5T^{2} \)
59 \( 1 - 512.T + 2.05e5T^{2} \)
61 \( 1 - 527. iT - 2.26e5T^{2} \)
67 \( 1 - 863.T + 3.00e5T^{2} \)
71 \( 1 - 810. iT - 3.57e5T^{2} \)
73 \( 1 + 157. iT - 3.89e5T^{2} \)
79 \( 1 - 796.T + 4.93e5T^{2} \)
83 \( 1 - 69.5T + 5.71e5T^{2} \)
89 \( 1 + 1.63e3iT - 7.04e5T^{2} \)
97 \( 1 - 904. iT - 9.12e5T^{2} \)
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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

−10.89990860290838448240169511491, −10.03737385537914793952315703128, −9.635147580454000534455052244731, −7.81884975615745781726973232631, −6.86326702476700258150113196485, −5.56409285141477231099783790725, −4.53603541389453771283699825542, −3.75394986958053568543166806119, −2.35078844318921428069062080928, −0.56751644935740964141343668151, 2.23896178067666223228653502375, 3.01453453683715713928464294288, 5.01027762790665390293750758906, 5.58185220359644124936355499772, 6.54823197114586738364337846327, 7.71959232544806552857610537059, 8.460155341715717213857930524257, 9.520439086257370532203801844613, 10.98395205473348751355055468120, 12.11581141192764893562146690843

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