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Properties

Label 2-312-104.77-c3-0-63
Degree $2$
Conductor $312$
Sign $0.818 + 0.575i$
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  + (2.82 + 0.121i)2-s + 3i·3-s + (7.97 + 0.689i)4-s − 14.7·5-s + (−0.365 + 8.47i)6-s − 28.6i·7-s + (22.4 + 2.92i)8-s − 9·9-s + (−41.7 − 1.80i)10-s + 55.0·11-s + (−2.06 + 23.9i)12-s + (21.7 − 41.5i)13-s + (3.49 − 80.8i)14-s − 44.2i·15-s + (63.0 + 10.9i)16-s − 111.·17-s + ⋯
L(s)  = 1  + (0.999 + 0.0431i)2-s + 0.577i·3-s + (0.996 + 0.0861i)4-s − 1.32·5-s + (−0.0249 + 0.576i)6-s − 1.54i·7-s + (0.991 + 0.129i)8-s − 0.333·9-s + (−1.31 − 0.0569i)10-s + 1.50·11-s + (−0.0497 + 0.575i)12-s + (0.464 − 0.885i)13-s + (0.0666 − 1.54i)14-s − 0.762i·15-s + (0.985 + 0.171i)16-s − 1.59·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.818 + 0.575i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.818 + 0.575i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(312\)    =    \(2^{3} \cdot 3 \cdot 13\)
Sign: $0.818 + 0.575i$
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: \((2,\ 312,\ (\ :3/2),\ 0.818 + 0.575i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.957803395\)
\(L(\frac12)\) \(\approx\) \(2.957803395\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.82 - 0.121i)T \)
3 \( 1 - 3iT \)
13 \( 1 + (-21.7 + 41.5i)T \)
good5 \( 1 + 14.7T + 125T^{2} \)
7 \( 1 + 28.6iT - 343T^{2} \)
11 \( 1 - 55.0T + 1.33e3T^{2} \)
17 \( 1 + 111.T + 4.91e3T^{2} \)
19 \( 1 - 80.7T + 6.85e3T^{2} \)
23 \( 1 - 137.T + 1.21e4T^{2} \)
29 \( 1 + 147. iT - 2.43e4T^{2} \)
31 \( 1 + 231. iT - 2.97e4T^{2} \)
37 \( 1 - 86.8T + 5.06e4T^{2} \)
41 \( 1 - 122. iT - 6.89e4T^{2} \)
43 \( 1 + 136. iT - 7.95e4T^{2} \)
47 \( 1 - 554. iT - 1.03e5T^{2} \)
53 \( 1 + 124. iT - 1.48e5T^{2} \)
59 \( 1 + 348.T + 2.05e5T^{2} \)
61 \( 1 - 674. iT - 2.26e5T^{2} \)
67 \( 1 - 156.T + 3.00e5T^{2} \)
71 \( 1 + 1.14e3iT - 3.57e5T^{2} \)
73 \( 1 + 775. iT - 3.89e5T^{2} \)
79 \( 1 + 1.10e3T + 4.93e5T^{2} \)
83 \( 1 - 446.T + 5.71e5T^{2} \)
89 \( 1 - 843. iT - 7.04e5T^{2} \)
97 \( 1 + 321. iT - 9.12e5T^{2} \)
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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

−11.22557699103257324893258901608, −10.71900468418610321456709980313, −9.359497574696749723270164680620, −7.953487777313275460239050215246, −7.21451711726308701151950641840, −6.23707066789787127468021050578, −4.53944991532120110279580670462, −4.09043565697459941337668268552, −3.24724963548181546815028043988, −0.861003205208538235224899855911, 1.56560533328119244801805246904, 3.01001824948246569112926020085, 4.10993911520610966720390384016, 5.25674155549496594302466176381, 6.59596005595822683883414155148, 7.02839901553140804072349803785, 8.531471923722006204066045759329, 9.101008227696795335688279127965, 11.19433134933597363475093280746, 11.55013949967701236281307856799

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