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Properties

Label 2-304-19.12-c4-0-4
Degree $2$
Conductor $304$
Sign $-0.660 - 0.750i$
Analytic cond. $31.4244$
Root an. cond. $5.60575$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−5.77 + 3.33i)3-s + (−1.87 − 3.25i)5-s + 40.7·7-s + (−18.2 + 31.6i)9-s + 49.6·11-s + (−234. − 135. i)13-s + (21.7 + 12.5i)15-s + (205. + 355. i)17-s + (351. + 81.3i)19-s + (−235. + 135. i)21-s + (185. − 320. i)23-s + (305. − 529. i)25-s − 783. i·27-s + (−526. − 303. i)29-s + 1.00e3i·31-s + ⋯
L(s)  = 1  + (−0.641 + 0.370i)3-s + (−0.0751 − 0.130i)5-s + 0.831·7-s + (−0.225 + 0.390i)9-s + 0.410·11-s + (−1.38 − 0.801i)13-s + (0.0964 + 0.0557i)15-s + (0.709 + 1.22i)17-s + (0.974 + 0.225i)19-s + (−0.533 + 0.307i)21-s + (0.349 − 0.606i)23-s + (0.488 − 0.846i)25-s − 1.07i·27-s + (−0.625 − 0.361i)29-s + 1.04i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.660 - 0.750i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.660 - 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $-0.660 - 0.750i$
Analytic conductor: \(31.4244\)
Root analytic conductor: \(5.60575\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 304,\ (\ :2),\ -0.660 - 0.750i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.9240275867\)
\(L(\frac12)\) \(\approx\) \(0.9240275867\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 + (-351. - 81.3i)T \)
good3 \( 1 + (5.77 - 3.33i)T + (40.5 - 70.1i)T^{2} \)
5 \( 1 + (1.87 + 3.25i)T + (-312.5 + 541. i)T^{2} \)
7 \( 1 - 40.7T + 2.40e3T^{2} \)
11 \( 1 - 49.6T + 1.46e4T^{2} \)
13 \( 1 + (234. + 135. i)T + (1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 + (-205. - 355. i)T + (-4.17e4 + 7.23e4i)T^{2} \)
23 \( 1 + (-185. + 320. i)T + (-1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (526. + 303. i)T + (3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 - 1.00e3iT - 9.23e5T^{2} \)
37 \( 1 - 2.45e3iT - 1.87e6T^{2} \)
41 \( 1 + (2.55e3 - 1.47e3i)T + (1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (-380. - 658. i)T + (-1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (1.73e3 - 3.00e3i)T + (-2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + (2.66e3 + 1.54e3i)T + (3.94e6 + 6.83e6i)T^{2} \)
59 \( 1 + (-498. + 288. i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (650. - 1.12e3i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (285. + 164. i)T + (1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + (4.87e3 - 2.81e3i)T + (1.27e7 - 2.20e7i)T^{2} \)
73 \( 1 + (-66.7 - 115. i)T + (-1.41e7 + 2.45e7i)T^{2} \)
79 \( 1 + (8.75e3 - 5.05e3i)T + (1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 + 1.72e3T + 4.74e7T^{2} \)
89 \( 1 + (-1.18e4 - 6.86e3i)T + (3.13e7 + 5.43e7i)T^{2} \)
97 \( 1 + (-5.16e3 + 2.97e3i)T + (4.42e7 - 7.66e7i)T^{2} \)
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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

−11.47116991941692570766793763982, −10.42880939254225212394107972999, −9.877586814037470230395365214360, −8.356346487623395554856237642634, −7.78556247319050250873584931787, −6.37559537378564044339342894412, −5.18872278423659068946373838013, −4.66831090700523876027783813044, −3.00923508611126761671156082966, −1.36096104206007078429905449441, 0.32333583860846028548030178331, 1.72379951487683329654278693183, 3.32303094572320566447362807180, 4.89337450538119171378376634205, 5.58819192930861245939507870182, 7.12783832631418007314173545515, 7.38779066733180935742133408568, 9.038295924055392446733203697378, 9.660761372464245233329991567933, 11.09952827906760252928439566104

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