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Properties

Label 2-312-104.77-c3-0-3
Degree $2$
Conductor $312$
Sign $-0.992 - 0.119i$
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  + (2.80 − 0.390i)2-s + 3i·3-s + (7.69 − 2.18i)4-s − 14.7·5-s + (1.17 + 8.40i)6-s − 0.217i·7-s + (20.6 − 9.14i)8-s − 9·9-s + (−41.4 + 5.78i)10-s − 43.4·11-s + (6.56 + 23.0i)12-s + (−23.9 + 40.2i)13-s + (−0.0851 − 0.609i)14-s − 44.3i·15-s + (54.4 − 33.6i)16-s − 16.1·17-s + ⋯
L(s)  = 1  + (0.990 − 0.138i)2-s + 0.577i·3-s + (0.961 − 0.273i)4-s − 1.32·5-s + (0.0797 + 0.571i)6-s − 0.0117i·7-s + (0.914 − 0.403i)8-s − 0.333·9-s + (−1.31 + 0.182i)10-s − 1.19·11-s + (0.158 + 0.555i)12-s + (−0.510 + 0.859i)13-s + (−0.00162 − 0.0116i)14-s − 0.763i·15-s + (0.850 − 0.526i)16-s − 0.230·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 - 0.119i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.992 - 0.119i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(312\)    =    \(2^{3} \cdot 3 \cdot 13\)
Sign: $-0.992 - 0.119i$
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: \((2,\ 312,\ (\ :3/2),\ -0.992 - 0.119i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.4865918874\)
\(L(\frac12)\) \(\approx\) \(0.4865918874\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.80 + 0.390i)T \)
3 \( 1 - 3iT \)
13 \( 1 + (23.9 - 40.2i)T \)
good5 \( 1 + 14.7T + 125T^{2} \)
7 \( 1 + 0.217iT - 343T^{2} \)
11 \( 1 + 43.4T + 1.33e3T^{2} \)
17 \( 1 + 16.1T + 4.91e3T^{2} \)
19 \( 1 + 72.7T + 6.85e3T^{2} \)
23 \( 1 + 83.4T + 1.21e4T^{2} \)
29 \( 1 + 128. iT - 2.43e4T^{2} \)
31 \( 1 - 306. iT - 2.97e4T^{2} \)
37 \( 1 + 340.T + 5.06e4T^{2} \)
41 \( 1 + 401. iT - 6.89e4T^{2} \)
43 \( 1 - 32.3iT - 7.95e4T^{2} \)
47 \( 1 + 118. iT - 1.03e5T^{2} \)
53 \( 1 - 486. iT - 1.48e5T^{2} \)
59 \( 1 - 241.T + 2.05e5T^{2} \)
61 \( 1 - 457. iT - 2.26e5T^{2} \)
67 \( 1 - 121.T + 3.00e5T^{2} \)
71 \( 1 + 251. iT - 3.57e5T^{2} \)
73 \( 1 + 66.4iT - 3.89e5T^{2} \)
79 \( 1 + 164.T + 4.93e5T^{2} \)
83 \( 1 - 311.T + 5.71e5T^{2} \)
89 \( 1 - 1.00e3iT - 7.04e5T^{2} \)
97 \( 1 + 433. iT - 9.12e5T^{2} \)
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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

−11.95122695062350058993543790867, −10.79936548952409557581244534656, −10.32335131525393440817835223081, −8.756358074101324365609216070236, −7.70813541432929599972885148537, −6.81602403683991170762683382235, −5.39929231951182122851024569595, −4.44554934733360292466231589157, −3.68614109256587282194325633010, −2.35836947927465205586263934305, 0.11557005335099657550168895234, 2.32518423693991638623638742992, 3.48676385687586487428467353269, 4.65274039811674551831408512002, 5.71141083666505819710122428781, 6.94629998305340678279329914167, 7.82217777308566188050932878190, 8.269869128131003227402052287144, 10.24865484095014834703059034885, 11.13308269811215087092146284664

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