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Properties

Label 2-312-104.77-c3-0-14
Degree $2$
Conductor $312$
Sign $-0.398 + 0.917i$
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  + (−0.457 + 2.79i)2-s + 3i·3-s + (−7.58 − 2.55i)4-s + 4.56·5-s + (−8.37 − 1.37i)6-s + 23.0i·7-s + (10.5 − 19.9i)8-s − 9·9-s + (−2.08 + 12.7i)10-s − 9.02·11-s + (7.65 − 22.7i)12-s + (3.63 + 46.7i)13-s + (−64.1 − 10.5i)14-s + 13.7i·15-s + (50.9 + 38.7i)16-s − 59.4·17-s + ⋯
L(s)  = 1  + (−0.161 + 0.986i)2-s + 0.577i·3-s + (−0.947 − 0.319i)4-s + 0.408·5-s + (−0.569 − 0.0933i)6-s + 1.24i·7-s + (0.468 − 0.883i)8-s − 0.333·9-s + (−0.0660 + 0.403i)10-s − 0.247·11-s + (0.184 − 0.547i)12-s + (0.0776 + 0.996i)13-s + (−1.22 − 0.200i)14-s + 0.235i·15-s + (0.796 + 0.604i)16-s − 0.847·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.398 + 0.917i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.398 + 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(312\)    =    \(2^{3} \cdot 3 \cdot 13\)
Sign: $-0.398 + 0.917i$
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: \((2,\ 312,\ (\ :3/2),\ -0.398 + 0.917i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.7130169423\)
\(L(\frac12)\) \(\approx\) \(0.7130169423\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.457 - 2.79i)T \)
3 \( 1 - 3iT \)
13 \( 1 + (-3.63 - 46.7i)T \)
good5 \( 1 - 4.56T + 125T^{2} \)
7 \( 1 - 23.0iT - 343T^{2} \)
11 \( 1 + 9.02T + 1.33e3T^{2} \)
17 \( 1 + 59.4T + 4.91e3T^{2} \)
19 \( 1 + 27.0T + 6.85e3T^{2} \)
23 \( 1 - 35.7T + 1.21e4T^{2} \)
29 \( 1 - 142. iT - 2.43e4T^{2} \)
31 \( 1 + 303. iT - 2.97e4T^{2} \)
37 \( 1 + 241.T + 5.06e4T^{2} \)
41 \( 1 + 280. iT - 6.89e4T^{2} \)
43 \( 1 - 198. iT - 7.95e4T^{2} \)
47 \( 1 + 319. iT - 1.03e5T^{2} \)
53 \( 1 - 531. iT - 1.48e5T^{2} \)
59 \( 1 - 378.T + 2.05e5T^{2} \)
61 \( 1 + 878. iT - 2.26e5T^{2} \)
67 \( 1 - 247.T + 3.00e5T^{2} \)
71 \( 1 + 182. iT - 3.57e5T^{2} \)
73 \( 1 - 815. iT - 3.89e5T^{2} \)
79 \( 1 + 817.T + 4.93e5T^{2} \)
83 \( 1 + 246.T + 5.71e5T^{2} \)
89 \( 1 - 355. iT - 7.04e5T^{2} \)
97 \( 1 - 506. iT - 9.12e5T^{2} \)
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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

−11.86341128190643033931918963455, −10.77643384248045134944369072696, −9.595513846489019043627249355772, −9.066658977127065715075133798932, −8.286208200466808726567463284906, −6.90302017674434340206138071117, −5.93937850517197410566779460718, −5.14459537211193363452742860209, −3.99379012740446711723007237151, −2.16510616306739711483879608532, 0.27457285293938218534573578656, 1.54344962316151436482886648253, 2.91897548479408206122947278407, 4.16792742517924334939511050684, 5.41409990397762997226500909972, 6.83903987545360642265736894603, 7.87305382654675183100319402881, 8.747109150715055653063470274764, 10.01814345507838914477183589048, 10.55925485277392536937518923627

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