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Properties

Label 2-304-19.12-c4-0-22
Degree $2$
Conductor $304$
Sign $0.686 + 0.727i$
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  + (−4.81 + 2.78i)3-s + (−13.2 − 22.9i)5-s + 84.4·7-s + (−25.0 + 43.3i)9-s − 33.3·11-s + (−32.2 − 18.5i)13-s + (127. + 73.7i)15-s + (−62.9 − 109. i)17-s + (354. + 69.2i)19-s + (−406. + 234. i)21-s + (−199. + 345. i)23-s + (−39.1 + 67.7i)25-s − 728. i·27-s + (−480. − 277. i)29-s + 1.32e3i·31-s + ⋯
L(s)  = 1  + (−0.535 + 0.308i)3-s + (−0.530 − 0.918i)5-s + 1.72·7-s + (−0.309 + 0.535i)9-s − 0.276·11-s + (−0.190 − 0.110i)13-s + (0.567 + 0.327i)15-s + (−0.217 − 0.377i)17-s + (0.981 + 0.191i)19-s + (−0.922 + 0.532i)21-s + (−0.376 + 0.652i)23-s + (−0.0625 + 0.108i)25-s − 0.999i·27-s + (−0.571 − 0.330i)29-s + 1.37i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.686 + 0.727i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.686 + 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $0.686 + 0.727i$
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: \((2,\ 304,\ (\ :2),\ 0.686 + 0.727i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.534947344\)
\(L(\frac12)\) \(\approx\) \(1.534947344\)
\(L(3)\) 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 \( 1 + (-354. - 69.2i)T \)
good3 \( 1 + (4.81 - 2.78i)T + (40.5 - 70.1i)T^{2} \)
5 \( 1 + (13.2 + 22.9i)T + (-312.5 + 541. i)T^{2} \)
7 \( 1 - 84.4T + 2.40e3T^{2} \)
11 \( 1 + 33.3T + 1.46e4T^{2} \)
13 \( 1 + (32.2 + 18.5i)T + (1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 + (62.9 + 109. i)T + (-4.17e4 + 7.23e4i)T^{2} \)
23 \( 1 + (199. - 345. i)T + (-1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (480. + 277. i)T + (3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 - 1.32e3iT - 9.23e5T^{2} \)
37 \( 1 + 630. iT - 1.87e6T^{2} \)
41 \( 1 + (-2.61e3 + 1.51e3i)T + (1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (953. + 1.65e3i)T + (-1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (-1.71e3 + 2.96e3i)T + (-2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + (527. + 304. i)T + (3.94e6 + 6.83e6i)T^{2} \)
59 \( 1 + (-4.22e3 + 2.44e3i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (-2.12e3 + 3.67e3i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (-3.19e3 - 1.84e3i)T + (1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + (588. - 339. i)T + (1.27e7 - 2.20e7i)T^{2} \)
73 \( 1 + (-2.27e3 - 3.94e3i)T + (-1.41e7 + 2.45e7i)T^{2} \)
79 \( 1 + (-6.77e3 + 3.91e3i)T + (1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 - 3.82e3T + 4.74e7T^{2} \)
89 \( 1 + (1.26e4 + 7.31e3i)T + (3.13e7 + 5.43e7i)T^{2} \)
97 \( 1 + (-8.15e3 + 4.70e3i)T + (4.42e7 - 7.66e7i)T^{2} \)
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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

−11.19344660859190980304903819473, −10.20987136918883500683411745390, −8.863637342173198869622433579248, −8.106364848194042990442246159528, −7.37233167610499644903206635335, −5.34263607066275543810130390345, −5.16134533811888955953033075129, −4.02946180663486464598934508470, −2.04588140792347282218719525200, −0.63175281834464038294200417117, 1.04345275606906075778323293300, 2.57957168517269379105863786986, 4.05973407053775493884978065032, 5.23419597325124075564028422914, 6.31358188850945357541484250852, 7.45160756690217645949369911097, 8.030929660314492865527326835688, 9.298014970052599807262615294347, 10.67087012529307761977542722777, 11.36074847807041055926828160163

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