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Properties

Label 2-304-304.211-c1-0-0
Degree $2$
Conductor $304$
Sign $-0.920 - 0.391i$
Analytic cond. $2.42745$
Root an. cond. $1.55802$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.143 + 1.40i)2-s + (−0.287 − 3.28i)3-s + (−1.95 + 0.404i)4-s + (−1.61 + 3.47i)5-s + (4.57 − 0.875i)6-s + (0.0566 + 0.0980i)7-s + (−0.850 − 2.69i)8-s + (−7.72 + 1.36i)9-s + (−5.12 − 1.77i)10-s + (−1.61 + 0.432i)11-s + (1.88 + 6.31i)12-s + (−0.319 + 3.64i)13-s + (−0.129 + 0.0937i)14-s + (11.8 + 4.31i)15-s + (3.67 − 1.58i)16-s + (−0.522 + 2.96i)17-s + ⋯
L(s)  = 1  + (0.101 + 0.994i)2-s + (−0.165 − 1.89i)3-s + (−0.979 + 0.202i)4-s + (−0.724 + 1.55i)5-s + (1.86 − 0.357i)6-s + (0.0213 + 0.0370i)7-s + (−0.300 − 0.953i)8-s + (−2.57 + 0.454i)9-s + (−1.61 − 0.562i)10-s + (−0.486 + 0.130i)11-s + (0.545 + 1.82i)12-s + (−0.0884 + 1.01i)13-s + (−0.0346 + 0.0250i)14-s + (3.06 + 1.11i)15-s + (0.918 − 0.396i)16-s + (−0.126 + 0.718i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.920 - 0.391i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.920 - 0.391i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $-0.920 - 0.391i$
Analytic conductor: \(2.42745\)
Root analytic conductor: \(1.55802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 304,\ (\ :1/2),\ -0.920 - 0.391i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0794538 + 0.389500i\)
\(L(\frac12)\) \(\approx\) \(0.0794538 + 0.389500i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.143 - 1.40i)T \)
19 \( 1 + (3.21 - 2.94i)T \)
good3 \( 1 + (0.287 + 3.28i)T + (-2.95 + 0.520i)T^{2} \)
5 \( 1 + (1.61 - 3.47i)T + (-3.21 - 3.83i)T^{2} \)
7 \( 1 + (-0.0566 - 0.0980i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.61 - 0.432i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (0.319 - 3.64i)T + (-12.8 - 2.25i)T^{2} \)
17 \( 1 + (0.522 - 2.96i)T + (-15.9 - 5.81i)T^{2} \)
23 \( 1 + (0.261 + 0.0953i)T + (17.6 + 14.7i)T^{2} \)
29 \( 1 + (-0.312 - 0.218i)T + (9.91 + 27.2i)T^{2} \)
31 \( 1 + (3.83 + 6.64i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.15 + 3.15i)T + 37iT^{2} \)
41 \( 1 + (-3.11 - 2.61i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (7.30 + 3.40i)T + (27.6 + 32.9i)T^{2} \)
47 \( 1 + (-5.19 + 0.915i)T + (44.1 - 16.0i)T^{2} \)
53 \( 1 + (-3.73 - 8.01i)T + (-34.0 + 40.6i)T^{2} \)
59 \( 1 + (0.301 + 0.430i)T + (-20.1 + 55.4i)T^{2} \)
61 \( 1 + (-1.15 - 2.48i)T + (-39.2 + 46.7i)T^{2} \)
67 \( 1 + (3.59 - 5.13i)T + (-22.9 - 62.9i)T^{2} \)
71 \( 1 + (-2.97 - 8.17i)T + (-54.3 + 45.6i)T^{2} \)
73 \( 1 + (-4.88 + 5.82i)T + (-12.6 - 71.8i)T^{2} \)
79 \( 1 + (-3.84 - 3.22i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (12.2 + 3.28i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + (-7.42 + 6.22i)T + (15.4 - 87.6i)T^{2} \)
97 \( 1 + (14.4 + 2.55i)T + (91.1 + 33.1i)T^{2} \)
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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

−12.26197803732622586170454572891, −11.47161633126902760326992737816, −10.42876491892178906139382244043, −8.709393236745795842019958966838, −7.81442549869916530174547526835, −7.21712117614602674168543786913, −6.55702468092482107019090094126, −5.79739037675339112739627198245, −3.83532956889872922529070575738, −2.28740128654106194427780287922, 0.27821717104333949578531614441, 3.07203876941240482003995479916, 4.13330340434692917880923631339, 4.95226509423102868698877071240, 5.41258729947362370385204013479, 8.184818554986605478755891204888, 8.812072822122312677885083955254, 9.526447456568251800362297307093, 10.48358567808628143160520429339, 11.14706201860780778771164637651

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