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Properties

Label 2-304-19.12-c4-0-12
Degree $2$
Conductor $304$
Sign $-0.511 - 0.859i$
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  + (−14.4 + 8.32i)3-s + (10.1 + 17.6i)5-s − 80.1·7-s + (98.0 − 169. i)9-s + 190.·11-s + (121. + 69.9i)13-s + (−293. − 169. i)15-s + (222. + 384. i)17-s + (330. + 144. i)19-s + (1.15e3 − 667. i)21-s + (291. − 505. i)23-s + (104. − 181. i)25-s + 1.91e3i·27-s + (−612. − 353. i)29-s + 491. i·31-s + ⋯
L(s)  = 1  + (−1.60 + 0.924i)3-s + (0.407 + 0.706i)5-s − 1.63·7-s + (1.21 − 2.09i)9-s + 1.57·11-s + (0.716 + 0.413i)13-s + (−1.30 − 0.754i)15-s + (0.768 + 1.33i)17-s + (0.916 + 0.400i)19-s + (2.62 − 1.51i)21-s + (0.551 − 0.955i)23-s + (0.167 − 0.289i)25-s + 2.63i·27-s + (−0.727 − 0.420i)29-s + 0.511i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.511 - 0.859i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.511 - 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $-0.511 - 0.859i$
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: \((2,\ 304,\ (\ :2),\ -0.511 - 0.859i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.066247420\)
\(L(\frac12)\) \(\approx\) \(1.066247420\)
\(L(3)\) 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 \( 1 + (-330. - 144. i)T \)
good3 \( 1 + (14.4 - 8.32i)T + (40.5 - 70.1i)T^{2} \)
5 \( 1 + (-10.1 - 17.6i)T + (-312.5 + 541. i)T^{2} \)
7 \( 1 + 80.1T + 2.40e3T^{2} \)
11 \( 1 - 190.T + 1.46e4T^{2} \)
13 \( 1 + (-121. - 69.9i)T + (1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 + (-222. - 384. i)T + (-4.17e4 + 7.23e4i)T^{2} \)
23 \( 1 + (-291. + 505. i)T + (-1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (612. + 353. i)T + (3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 - 491. iT - 9.23e5T^{2} \)
37 \( 1 + 1.36e3iT - 1.87e6T^{2} \)
41 \( 1 + (-215. + 124. i)T + (1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (-1.04e3 - 1.81e3i)T + (-1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (363. - 630. i)T + (-2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + (718. + 415. i)T + (3.94e6 + 6.83e6i)T^{2} \)
59 \( 1 + (-2.63e3 + 1.52e3i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (2.84e3 - 4.93e3i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (738. + 426. i)T + (1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + (1.75e3 - 1.01e3i)T + (1.27e7 - 2.20e7i)T^{2} \)
73 \( 1 + (-4.24e3 - 7.34e3i)T + (-1.41e7 + 2.45e7i)T^{2} \)
79 \( 1 + (-6.52e3 + 3.76e3i)T + (1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 - 1.66e3T + 4.74e7T^{2} \)
89 \( 1 + (3.82e3 + 2.20e3i)T + (3.13e7 + 5.43e7i)T^{2} \)
97 \( 1 + (6.16e3 - 3.56e3i)T + (4.42e7 - 7.66e7i)T^{2} \)
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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

−11.23282893792607381828807660076, −10.44782563068881561415287678504, −9.773935996301762904002966325667, −9.087285686049812465177832924312, −6.88483558826915791024734022845, −6.26204600877435641123570816969, −5.81837494369864060671521724464, −4.11667771796063177607919682455, −3.42890429074853381530275684279, −1.01019094322636373960036347770, 0.58565248030952884372972153748, 1.31036330151159251098765738799, 3.41019586431938484826995888230, 5.12773084399967881724515552257, 5.86421944418213914465478762953, 6.69355638029569183424878732706, 7.39086953140300708254486705292, 9.210053471980591930639299454009, 9.715429467953229563910886537938, 11.06139663542776527049623636136

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