Properties

Label 2-304-19.12-c4-0-1
Degree $2$
Conductor $304$
Sign $-0.574 - 0.818i$
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  + (9.05 − 5.22i)3-s + (−12.7 − 22.0i)5-s − 27.7·7-s + (14.1 − 24.4i)9-s − 143.·11-s + (123. + 71.5i)13-s + (−230. − 133. i)15-s + (43.9 + 76.1i)17-s + (−340. − 119. i)19-s + (−251. + 145. i)21-s + (−350. + 607. i)23-s + (−12.6 + 21.9i)25-s + 551. i·27-s + (−37.5 − 21.6i)29-s + 111. i·31-s + ⋯
L(s)  = 1  + (1.00 − 0.580i)3-s + (−0.510 − 0.883i)5-s − 0.566·7-s + (0.174 − 0.301i)9-s − 1.18·11-s + (0.733 + 0.423i)13-s + (−1.02 − 0.592i)15-s + (0.152 + 0.263i)17-s + (−0.943 − 0.331i)19-s + (−0.569 + 0.328i)21-s + (−0.663 + 1.14i)23-s + (−0.0202 + 0.0350i)25-s + 0.756i·27-s + (−0.0446 − 0.0257i)29-s + 0.115i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.574 - 0.818i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.574 - 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $-0.574 - 0.818i$
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: \((2,\ 304,\ (\ :2),\ -0.574 - 0.818i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.1471772627\)
\(L(\frac12)\) \(\approx\) \(0.1471772627\)
\(L(3)\) 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 \( 1 + (340. + 119. i)T \)
good3 \( 1 + (-9.05 + 5.22i)T + (40.5 - 70.1i)T^{2} \)
5 \( 1 + (12.7 + 22.0i)T + (-312.5 + 541. i)T^{2} \)
7 \( 1 + 27.7T + 2.40e3T^{2} \)
11 \( 1 + 143.T + 1.46e4T^{2} \)
13 \( 1 + (-123. - 71.5i)T + (1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 + (-43.9 - 76.1i)T + (-4.17e4 + 7.23e4i)T^{2} \)
23 \( 1 + (350. - 607. i)T + (-1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (37.5 + 21.6i)T + (3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 - 111. iT - 9.23e5T^{2} \)
37 \( 1 - 1.07e3iT - 1.87e6T^{2} \)
41 \( 1 + (-80.1 + 46.2i)T + (1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (945. + 1.63e3i)T + (-1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (1.36e3 - 2.36e3i)T + (-2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + (-2.15e3 - 1.24e3i)T + (3.94e6 + 6.83e6i)T^{2} \)
59 \( 1 + (-5.03e3 + 2.90e3i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (-1.26e3 + 2.18e3i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (4.30e3 + 2.48e3i)T + (1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + (3.72e3 - 2.14e3i)T + (1.27e7 - 2.20e7i)T^{2} \)
73 \( 1 + (1.58e3 + 2.75e3i)T + (-1.41e7 + 2.45e7i)T^{2} \)
79 \( 1 + (4.93e3 - 2.84e3i)T + (1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 + 1.04e4T + 4.74e7T^{2} \)
89 \( 1 + (-4.82e3 - 2.78e3i)T + (3.13e7 + 5.43e7i)T^{2} \)
97 \( 1 + (1.56e4 - 9.01e3i)T + (4.42e7 - 7.66e7i)T^{2} \)
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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

−11.53096308362196087304303056028, −10.38498716577438105875755639666, −9.241586300638256990060463481313, −8.362513399022586347930649830026, −7.920250108117287799318991954129, −6.71745813932536992999858778286, −5.38298167674011996229397697774, −4.06767937645734103561754418800, −2.89167311818885046019838377899, −1.60104578628561612191360973522, 0.03690702347921764801719775841, 2.50396130143368765249354424152, 3.27382956443226548626904796048, 4.21420846632728522007002931423, 5.83050906681421204059549638139, 6.97162742801102792637609313304, 8.096682935469591251632237459070, 8.694131034950907692747770451664, 10.03665410690345655702285894577, 10.44585169586109092821584892774

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