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Properties

Label 2-304-304.179-c1-0-34
Degree $2$
Conductor $304$
Sign $-0.995 - 0.0958i$
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  + (1.04 − 0.952i)2-s + (−2.47 + 0.662i)3-s + (0.184 − 1.99i)4-s + (0.0905 + 0.338i)5-s + (−1.95 + 3.04i)6-s − 1.90·7-s + (−1.70 − 2.25i)8-s + (3.08 − 1.77i)9-s + (0.416 + 0.266i)10-s + (−3.26 − 3.26i)11-s + (0.864 + 5.04i)12-s + (−2.55 − 0.685i)13-s + (−1.98 + 1.81i)14-s + (−0.448 − 0.776i)15-s + (−3.93 − 0.733i)16-s + (−2.56 + 4.43i)17-s + ⋯
L(s)  = 1  + (0.738 − 0.673i)2-s + (−1.42 + 0.382i)3-s + (0.0921 − 0.995i)4-s + (0.0405 + 0.151i)5-s + (−0.797 + 1.24i)6-s − 0.718·7-s + (−0.602 − 0.797i)8-s + (1.02 − 0.592i)9-s + (0.131 + 0.0844i)10-s + (−0.984 − 0.984i)11-s + (0.249 + 1.45i)12-s + (−0.709 − 0.189i)13-s + (−0.530 + 0.483i)14-s + (−0.115 − 0.200i)15-s + (−0.983 − 0.183i)16-s + (−0.621 + 1.07i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0958i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 - 0.0958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $-0.995 - 0.0958i$
Analytic conductor: \(2.42745\)
Root analytic conductor: \(1.55802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 304,\ (\ :1/2),\ -0.995 - 0.0958i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0189558 + 0.394772i\)
\(L(\frac12)\) \(\approx\) \(0.0189558 + 0.394772i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.04 + 0.952i)T \)
19 \( 1 + (4.32 - 0.528i)T \)
good3 \( 1 + (2.47 - 0.662i)T + (2.59 - 1.5i)T^{2} \)
5 \( 1 + (-0.0905 - 0.338i)T + (-4.33 + 2.5i)T^{2} \)
7 \( 1 + 1.90T + 7T^{2} \)
11 \( 1 + (3.26 + 3.26i)T + 11iT^{2} \)
13 \( 1 + (2.55 + 0.685i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (2.56 - 4.43i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (0.732 + 1.26i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.01 - 1.07i)T + (25.1 + 14.5i)T^{2} \)
31 \( 1 - 6.04T + 31T^{2} \)
37 \( 1 + (7.05 + 7.05i)T + 37iT^{2} \)
41 \( 1 + (-4.34 + 7.52i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.39 + 1.17i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (5.48 - 3.16i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.475 + 1.77i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-0.0988 - 0.368i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (0.305 - 1.13i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (1.64 - 6.13i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-4.43 - 2.55i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-5.92 - 3.41i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.48 + 7.76i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.502 + 0.502i)T - 83iT^{2} \)
89 \( 1 + (0.419 + 0.726i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.939 + 0.542i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.01595818546416920632691033144, −10.65219803454763752024398323787, −10.01570158793726068557977212141, −8.572352438081198865356073957472, −6.69769819546651492636601602594, −6.04034943812079493156935129475, −5.16654545601595409747309437583, −4.11698771537866651041709484813, −2.65874108838782461894459405828, −0.25056705059946253127854410268, 2.67526908328129428987181516788, 4.68551416971661388707899543659, 5.10227317457054532052550352451, 6.48824703354904148197402225056, 6.83664672218489976116734251053, 7.975719002255710213161781429081, 9.453881549258635366591851120454, 10.56447175318100792697514188105, 11.61525433329221903994096432573, 12.36766503559316405686611621501

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