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Properties

Label 2-312-104.77-c3-0-22
Degree $2$
Conductor $312$
Sign $0.250 - 0.968i$
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  + (−2.81 − 0.274i)2-s + 3i·3-s + (7.84 + 1.54i)4-s − 8.71·5-s + (0.824 − 8.44i)6-s − 16.6i·7-s + (−21.6 − 6.51i)8-s − 9·9-s + (24.5 + 2.39i)10-s − 35.9·11-s + (−4.64 + 23.5i)12-s + (46.8 − 1.83i)13-s + (−4.58 + 46.9i)14-s − 26.1i·15-s + (59.2 + 24.2i)16-s + 23.1·17-s + ⋯
L(s)  = 1  + (−0.995 − 0.0971i)2-s + 0.577i·3-s + (0.981 + 0.193i)4-s − 0.779·5-s + (0.0560 − 0.574i)6-s − 0.899i·7-s + (−0.957 − 0.287i)8-s − 0.333·9-s + (0.775 + 0.0757i)10-s − 0.985·11-s + (−0.111 + 0.566i)12-s + (0.999 − 0.0392i)13-s + (−0.0874 + 0.895i)14-s − 0.450i·15-s + (0.925 + 0.379i)16-s + 0.330·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.250 - 0.968i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.250 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(312\)    =    \(2^{3} \cdot 3 \cdot 13\)
Sign: $0.250 - 0.968i$
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: \((2,\ 312,\ (\ :3/2),\ 0.250 - 0.968i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.7233917465\)
\(L(\frac12)\) \(\approx\) \(0.7233917465\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.81 + 0.274i)T \)
3 \( 1 - 3iT \)
13 \( 1 + (-46.8 + 1.83i)T \)
good5 \( 1 + 8.71T + 125T^{2} \)
7 \( 1 + 16.6iT - 343T^{2} \)
11 \( 1 + 35.9T + 1.33e3T^{2} \)
17 \( 1 - 23.1T + 4.91e3T^{2} \)
19 \( 1 - 30.2T + 6.85e3T^{2} \)
23 \( 1 - 72.9T + 1.21e4T^{2} \)
29 \( 1 - 119. iT - 2.43e4T^{2} \)
31 \( 1 - 96.3iT - 2.97e4T^{2} \)
37 \( 1 + 33.6T + 5.06e4T^{2} \)
41 \( 1 + 60.2iT - 6.89e4T^{2} \)
43 \( 1 + 6.93iT - 7.95e4T^{2} \)
47 \( 1 - 452. iT - 1.03e5T^{2} \)
53 \( 1 - 308. iT - 1.48e5T^{2} \)
59 \( 1 - 92.9T + 2.05e5T^{2} \)
61 \( 1 - 538. iT - 2.26e5T^{2} \)
67 \( 1 - 484.T + 3.00e5T^{2} \)
71 \( 1 - 447. iT - 3.57e5T^{2} \)
73 \( 1 - 763. iT - 3.89e5T^{2} \)
79 \( 1 - 354.T + 4.93e5T^{2} \)
83 \( 1 + 232.T + 5.71e5T^{2} \)
89 \( 1 - 1.31e3iT - 7.04e5T^{2} \)
97 \( 1 - 1.33e3iT - 9.12e5T^{2} \)
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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

−10.95673467676045585229285284442, −10.66719278581220763827302905339, −9.636860969251879299106716987755, −8.562922349116070354793611043013, −7.79378910709490481440840609221, −6.95264114034795948551947459807, −5.53211853719303259558683578484, −4.01297662320083833710811422273, −3.01155174402872136269377904080, −0.998805556098292689700177974441, 0.46285098012106068700911004779, 2.08742294182894605063830047604, 3.34045602431170568087551565846, 5.37311149789673425737128064756, 6.32312730891420545831423894117, 7.51306319717005783362337957816, 8.157867656215473669372942217397, 8.914033128086512888030796725492, 10.05274612496076816654442212425, 11.18122013690879023038178561089

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