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Properties

Label 2-304-19.9-c1-0-7
Degree $2$
Conductor $304$
Sign $0.363 + 0.931i$
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  + (2.20 − 1.85i)3-s + (−0.826 − 0.300i)5-s + (0.173 − 0.300i)7-s + (0.918 − 5.21i)9-s + (−1.11 − 1.92i)11-s + (1.97 + 1.65i)13-s + (−2.37 + 0.866i)15-s + (0.0812 + 0.460i)17-s + (4.29 + 0.725i)19-s + (−0.173 − 0.984i)21-s + (−2.53 + 0.921i)23-s + (−3.23 − 2.71i)25-s + (−3.29 − 5.71i)27-s + (−1.19 + 6.77i)29-s + (−3.55 + 6.15i)31-s + ⋯
L(s)  = 1  + (1.27 − 1.06i)3-s + (−0.369 − 0.134i)5-s + (0.0656 − 0.113i)7-s + (0.306 − 1.73i)9-s + (−0.335 − 0.581i)11-s + (0.546 + 0.458i)13-s + (−0.614 + 0.223i)15-s + (0.0197 + 0.111i)17-s + (0.986 + 0.166i)19-s + (−0.0378 − 0.214i)21-s + (−0.527 + 0.192i)23-s + (−0.647 − 0.543i)25-s + (−0.634 − 1.09i)27-s + (−0.221 + 1.25i)29-s + (−0.638 + 1.10i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.363 + 0.931i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.363 + 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $0.363 + 0.931i$
Analytic conductor: \(2.42745\)
Root analytic conductor: \(1.55802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 304,\ (\ :1/2),\ 0.363 + 0.931i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.49089 - 1.01840i\)
\(L(\frac12)\) \(\approx\) \(1.49089 - 1.01840i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 + (-4.29 - 0.725i)T \)
good3 \( 1 + (-2.20 + 1.85i)T + (0.520 - 2.95i)T^{2} \)
5 \( 1 + (0.826 + 0.300i)T + (3.83 + 3.21i)T^{2} \)
7 \( 1 + (-0.173 + 0.300i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.11 + 1.92i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.97 - 1.65i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (-0.0812 - 0.460i)T + (-15.9 + 5.81i)T^{2} \)
23 \( 1 + (2.53 - 0.921i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (1.19 - 6.77i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (3.55 - 6.15i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 4.94T + 37T^{2} \)
41 \( 1 + (-1.89 + 1.59i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-3.66 - 1.33i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-1.26 + 7.18i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (-2.66 + 0.970i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (-1.09 - 6.20i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (8.57 - 3.12i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (1.33 - 7.55i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (8.74 + 3.18i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (-1.06 + 0.892i)T + (12.6 - 71.8i)T^{2} \)
79 \( 1 + (-9.07 + 7.61i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (7.41 - 12.8i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (7.88 + 6.61i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (-1.64 - 9.30i)T + (-91.1 + 33.1i)T^{2} \)
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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

−11.80231081361671690412324919942, −10.63737459992635744672454839646, −9.285258717950057319187386568039, −8.562465699837493159834469270340, −7.75462241733897391334685503359, −7.01600077234049932217550653133, −5.73639091453123901622686686063, −3.94641791643319593983671862706, −2.89830795757615260192454112119, −1.42098548456252627329797119248, 2.41367861495872853687075273129, 3.57496162976818016502204048641, 4.44707007143136812035931794302, 5.75882909421681805225593036056, 7.57242280431315179408913772745, 8.052171051097653478418956569147, 9.277507864517465210965796358642, 9.771676579953384422739818291192, 10.78557163843365360329171817323, 11.73208295835311959405004247945

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