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Properties

Label 2-312-104.77-c3-0-33
Degree $2$
Conductor $312$
Sign $0.859 - 0.510i$
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  + (−2.09 − 1.90i)2-s + 3i·3-s + (0.748 + 7.96i)4-s + 5.98·5-s + (5.71 − 6.27i)6-s + 13.5i·7-s + (13.5 − 18.0i)8-s − 9·9-s + (−12.5 − 11.3i)10-s + 38.8·11-s + (−23.8 + 2.24i)12-s + (17.8 − 43.3i)13-s + (25.8 − 28.3i)14-s + 17.9i·15-s + (−62.8 + 11.9i)16-s + 10.7·17-s + ⋯
L(s)  = 1  + (−0.739 − 0.673i)2-s + 0.577i·3-s + (0.0935 + 0.995i)4-s + 0.535·5-s + (0.388 − 0.426i)6-s + 0.732i·7-s + (0.601 − 0.799i)8-s − 0.333·9-s + (−0.395 − 0.360i)10-s + 1.06·11-s + (−0.574 + 0.0540i)12-s + (0.380 − 0.924i)13-s + (0.493 − 0.541i)14-s + 0.308i·15-s + (−0.982 + 0.186i)16-s + 0.153·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 - 0.510i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.859 - 0.510i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(312\)    =    \(2^{3} \cdot 3 \cdot 13\)
Sign: $0.859 - 0.510i$
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: \((2,\ 312,\ (\ :3/2),\ 0.859 - 0.510i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.469489644\)
\(L(\frac12)\) \(\approx\) \(1.469489644\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.09 + 1.90i)T \)
3 \( 1 - 3iT \)
13 \( 1 + (-17.8 + 43.3i)T \)
good5 \( 1 - 5.98T + 125T^{2} \)
7 \( 1 - 13.5iT - 343T^{2} \)
11 \( 1 - 38.8T + 1.33e3T^{2} \)
17 \( 1 - 10.7T + 4.91e3T^{2} \)
19 \( 1 - 50.0T + 6.85e3T^{2} \)
23 \( 1 - 58.0T + 1.21e4T^{2} \)
29 \( 1 - 140. iT - 2.43e4T^{2} \)
31 \( 1 - 49.2iT - 2.97e4T^{2} \)
37 \( 1 - 109.T + 5.06e4T^{2} \)
41 \( 1 - 200. iT - 6.89e4T^{2} \)
43 \( 1 - 53.5iT - 7.95e4T^{2} \)
47 \( 1 - 95.3iT - 1.03e5T^{2} \)
53 \( 1 - 385. iT - 1.48e5T^{2} \)
59 \( 1 - 305.T + 2.05e5T^{2} \)
61 \( 1 - 164. iT - 2.26e5T^{2} \)
67 \( 1 - 962.T + 3.00e5T^{2} \)
71 \( 1 + 195. iT - 3.57e5T^{2} \)
73 \( 1 + 317. iT - 3.89e5T^{2} \)
79 \( 1 + 221.T + 4.93e5T^{2} \)
83 \( 1 - 445.T + 5.71e5T^{2} \)
89 \( 1 - 523. iT - 7.04e5T^{2} \)
97 \( 1 - 147. iT - 9.12e5T^{2} \)
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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

−11.21134155262245655947271730088, −10.29240086876063117644557363074, −9.431831510962932626652739838940, −8.879474285033143345829764150546, −7.82050673187444742489914277715, −6.44055748069644478768753340046, −5.27591898346481411366751168289, −3.78756435599315693963559721972, −2.71747755373699698729287805497, −1.20755407090072979891103423082, 0.839364575551442153096040475399, 1.95549895716410814754300699083, 4.05680582004683615166751939017, 5.56886203315045972951755443732, 6.55160106545179815221942918544, 7.17342835573860584688802977576, 8.257395421369844316372558973403, 9.277368849927655830647621461775, 9.916512162522840508832480469761, 11.13120974560011646303339402236

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