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Properties

Label 2-304-19.8-c4-0-31
Degree $2$
Conductor $304$
Sign $-0.163 + 0.986i$
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  + (−2.52 − 1.45i)3-s + (22.1 − 38.3i)5-s + 11.1·7-s + (−36.2 − 62.7i)9-s + 221.·11-s + (177. − 102. i)13-s + (−111. + 64.5i)15-s + (−201. + 348. i)17-s + (255. − 255. i)19-s + (−28.2 − 16.3i)21-s + (76.8 + 133. i)23-s + (−666. − 1.15e3i)25-s + 447. i·27-s + (510. − 294. i)29-s + 623. i·31-s + ⋯
L(s)  = 1  + (−0.280 − 0.162i)3-s + (0.884 − 1.53i)5-s + 0.228·7-s + (−0.447 − 0.775i)9-s + 1.82·11-s + (1.05 − 0.606i)13-s + (−0.496 + 0.286i)15-s + (−0.697 + 1.20i)17-s + (0.707 − 0.706i)19-s + (−0.0640 − 0.0370i)21-s + (0.145 + 0.251i)23-s + (−1.06 − 1.84i)25-s + 0.614i·27-s + (0.607 − 0.350i)29-s + 0.648i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.163 + 0.986i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.163 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $-0.163 + 0.986i$
Analytic conductor: \(31.4244\)
Root analytic conductor: \(5.60575\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 304,\ (\ :2),\ -0.163 + 0.986i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.406499187\)
\(L(\frac12)\) \(\approx\) \(2.406499187\)
\(L(3)\) 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 \( 1 + (-255. + 255. i)T \)
good3 \( 1 + (2.52 + 1.45i)T + (40.5 + 70.1i)T^{2} \)
5 \( 1 + (-22.1 + 38.3i)T + (-312.5 - 541. i)T^{2} \)
7 \( 1 - 11.1T + 2.40e3T^{2} \)
11 \( 1 - 221.T + 1.46e4T^{2} \)
13 \( 1 + (-177. + 102. i)T + (1.42e4 - 2.47e4i)T^{2} \)
17 \( 1 + (201. - 348. i)T + (-4.17e4 - 7.23e4i)T^{2} \)
23 \( 1 + (-76.8 - 133. i)T + (-1.39e5 + 2.42e5i)T^{2} \)
29 \( 1 + (-510. + 294. i)T + (3.53e5 - 6.12e5i)T^{2} \)
31 \( 1 - 623. iT - 9.23e5T^{2} \)
37 \( 1 + 776. iT - 1.87e6T^{2} \)
41 \( 1 + (-58.6 - 33.8i)T + (1.41e6 + 2.44e6i)T^{2} \)
43 \( 1 + (1.02e3 - 1.78e3i)T + (-1.70e6 - 2.96e6i)T^{2} \)
47 \( 1 + (-875. - 1.51e3i)T + (-2.43e6 + 4.22e6i)T^{2} \)
53 \( 1 + (-3.95e3 + 2.28e3i)T + (3.94e6 - 6.83e6i)T^{2} \)
59 \( 1 + (5.56e3 + 3.21e3i)T + (6.05e6 + 1.04e7i)T^{2} \)
61 \( 1 + (-286. - 495. i)T + (-6.92e6 + 1.19e7i)T^{2} \)
67 \( 1 + (-4.02e3 + 2.32e3i)T + (1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 + (5.65e3 + 3.26e3i)T + (1.27e7 + 2.20e7i)T^{2} \)
73 \( 1 + (-5.19e3 + 9.00e3i)T + (-1.41e7 - 2.45e7i)T^{2} \)
79 \( 1 + (-1.11e3 - 642. i)T + (1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 + 4.09e3T + 4.74e7T^{2} \)
89 \( 1 + (8.24e3 - 4.76e3i)T + (3.13e7 - 5.43e7i)T^{2} \)
97 \( 1 + (-2.73e3 - 1.58e3i)T + (4.42e7 + 7.66e7i)T^{2} \)
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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

−10.97522005928120204939568656107, −9.542436757222267866823608377413, −8.981018720841156487226551350743, −8.319994837135258800905627326522, −6.49041618756587963535369442577, −5.97548444214539675793188037303, −4.80632861091105576128676860535, −3.64085967545468782436060330447, −1.54688400580987601315525544872, −0.865245155728816248326155370780, 1.55007143395697598832389261153, 2.79455539942626497024035811645, 4.07406475727481762659046064779, 5.58501628248075545131525869313, 6.48895061812806947027274785973, 7.12045179011806536614011025365, 8.658127210521459437717343630016, 9.588299151457351519790780288412, 10.49479466098467638442865909378, 11.38554963082053504668485007470

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