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Properties

Label 2-304-19.12-c4-0-5
Degree $2$
Conductor $304$
Sign $-0.965 - 0.260i$
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  + (−7.37 + 4.25i)3-s + (7.14 + 12.3i)5-s + 35.7·7-s + (−4.26 + 7.37i)9-s − 184.·11-s + (245. + 141. i)13-s + (−105. − 60.8i)15-s + (53.1 + 92.1i)17-s + (338. − 126. i)19-s + (−263. + 152. i)21-s + (−412. + 713. i)23-s + (210. − 364. i)25-s − 762. i·27-s + (875. + 505. i)29-s + 370. i·31-s + ⋯
L(s)  = 1  + (−0.819 + 0.472i)3-s + (0.285 + 0.494i)5-s + 0.728·7-s + (−0.0525 + 0.0910i)9-s − 1.52·11-s + (1.45 + 0.840i)13-s + (−0.468 − 0.270i)15-s + (0.184 + 0.318i)17-s + (0.936 − 0.350i)19-s + (−0.597 + 0.344i)21-s + (−0.779 + 1.34i)23-s + (0.336 − 0.583i)25-s − 1.04i·27-s + (1.04 + 0.601i)29-s + 0.385i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 - 0.260i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.965 - 0.260i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $-0.965 - 0.260i$
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: \((2,\ 304,\ (\ :2),\ -0.965 - 0.260i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.9657210682\)
\(L(\frac12)\) \(\approx\) \(0.9657210682\)
\(L(3)\) 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 \( 1 + (-338. + 126. i)T \)
good3 \( 1 + (7.37 - 4.25i)T + (40.5 - 70.1i)T^{2} \)
5 \( 1 + (-7.14 - 12.3i)T + (-312.5 + 541. i)T^{2} \)
7 \( 1 - 35.7T + 2.40e3T^{2} \)
11 \( 1 + 184.T + 1.46e4T^{2} \)
13 \( 1 + (-245. - 141. i)T + (1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 + (-53.1 - 92.1i)T + (-4.17e4 + 7.23e4i)T^{2} \)
23 \( 1 + (412. - 713. i)T + (-1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (-875. - 505. i)T + (3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 - 370. iT - 9.23e5T^{2} \)
37 \( 1 + 235. iT - 1.87e6T^{2} \)
41 \( 1 + (2.25e3 - 1.29e3i)T + (1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (1.62e3 + 2.82e3i)T + (-1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (1.51e3 - 2.61e3i)T + (-2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + (497. + 287. i)T + (3.94e6 + 6.83e6i)T^{2} \)
59 \( 1 + (4.60e3 - 2.65e3i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (1.97e3 - 3.41e3i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (-148. - 85.9i)T + (1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + (-1.60e3 + 927. i)T + (1.27e7 - 2.20e7i)T^{2} \)
73 \( 1 + (4.13e3 + 7.15e3i)T + (-1.41e7 + 2.45e7i)T^{2} \)
79 \( 1 + (1.68e3 - 975. i)T + (1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 - 1.01e4T + 4.74e7T^{2} \)
89 \( 1 + (1.01e4 + 5.87e3i)T + (3.13e7 + 5.43e7i)T^{2} \)
97 \( 1 + (8.45e3 - 4.88e3i)T + (4.42e7 - 7.66e7i)T^{2} \)
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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

−11.29085656794957760760853581161, −10.67749258235064323703701445084, −9.999585933563351470085990784843, −8.589126809720852820928143410026, −7.72466989786519601617757494693, −6.40104293410950463461545367118, −5.46854724979185829370898918523, −4.67236322280329304236457743420, −3.14742688335668456676607331458, −1.58529392647944079663027891368, 0.34339364019531295803983660647, 1.44546188613385195277190676808, 3.13940242176078098962553726312, 4.91635393814741582143014921830, 5.57435120014590178948452311616, 6.51423715390563879509823788183, 7.949861023771416501077168311979, 8.431364003840116085403100120920, 9.918689981267792924667672357183, 10.79829962600473281473653304079

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