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Properties

Label 2-304-19.11-c3-0-18
Degree $2$
Conductor $304$
Sign $0.0656 + 0.997i$
Analytic cond. $17.9365$
Root an. cond. $4.23516$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.16 + 5.48i)3-s + (−2.96 + 5.13i)5-s − 0.660·7-s + (−6.59 − 11.4i)9-s − 50.1·11-s + (−1.31 − 2.27i)13-s + (−18.7 − 32.5i)15-s + (−28.9 + 50.0i)17-s + (12.1 − 81.9i)19-s + (2.09 − 3.62i)21-s + (18.9 + 32.8i)23-s + (44.9 + 77.7i)25-s − 87.5·27-s + (−154. − 268. i)29-s + 282.·31-s + ⋯
L(s)  = 1  + (−0.609 + 1.05i)3-s + (−0.265 + 0.459i)5-s − 0.0356·7-s + (−0.244 − 0.422i)9-s − 1.37·11-s + (−0.0280 − 0.0485i)13-s + (−0.323 − 0.560i)15-s + (−0.412 + 0.714i)17-s + (0.146 − 0.989i)19-s + (0.0217 − 0.0377i)21-s + (0.172 + 0.298i)23-s + (0.359 + 0.622i)25-s − 0.624·27-s + (−0.991 − 1.71i)29-s + 1.63·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0656 + 0.997i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0656 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $0.0656 + 0.997i$
Analytic conductor: \(17.9365\)
Root analytic conductor: \(4.23516\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 304,\ (\ :3/2),\ 0.0656 + 0.997i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.1727850087\)
\(L(\frac12)\) \(\approx\) \(0.1727850087\)
\(L(\frac{5}{2})\) 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 \( 1 + (-12.1 + 81.9i)T \)
good3 \( 1 + (3.16 - 5.48i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + (2.96 - 5.13i)T + (-62.5 - 108. i)T^{2} \)
7 \( 1 + 0.660T + 343T^{2} \)
11 \( 1 + 50.1T + 1.33e3T^{2} \)
13 \( 1 + (1.31 + 2.27i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (28.9 - 50.0i)T + (-2.45e3 - 4.25e3i)T^{2} \)
23 \( 1 + (-18.9 - 32.8i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (154. + 268. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 - 282.T + 2.97e4T^{2} \)
37 \( 1 + 21.0T + 5.06e4T^{2} \)
41 \( 1 + (-172. + 297. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-132. + 229. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (69.8 + 121. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-66.8 - 115. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-227. + 394. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (54.2 + 93.9i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-13.1 - 22.8i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (559. - 969. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + (-235. + 407. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-190. + 330. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 1.32e3T + 5.71e5T^{2} \)
89 \( 1 + (202. + 351. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (687. - 1.19e3i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−10.95396684792483281357979674056, −10.30712097785854261883430099589, −9.457565068237412439729111577049, −8.186042850889176312859598109004, −7.17633973649647548287138987592, −5.84791268403458625566117901486, −4.95874462500650072290888289891, −3.92998007852340956214441459856, −2.55197351874660565564436147643, −0.07397464458509764465358505220, 1.23843128283815124696626143525, 2.80207263104132668283034144502, 4.60288303543668611199571119799, 5.62220529136947452820813405784, 6.67287903662604757643520502509, 7.62586450922129942155809375051, 8.378190178357731183697963166509, 9.685439132800896080905886944280, 10.77700504311051966099156475195, 11.66069813325822913097950217236

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