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Properties

Label 2-304-304.179-c1-0-35
Degree $2$
Conductor $304$
Sign $-0.589 + 0.807i$
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  + (−0.180 − 1.40i)2-s + (3.14 − 0.844i)3-s + (−1.93 + 0.505i)4-s + (−0.521 − 1.94i)5-s + (−1.75 − 4.26i)6-s − 3.12·7-s + (1.05 + 2.62i)8-s + (6.61 − 3.81i)9-s + (−2.63 + 1.08i)10-s + (−0.956 − 0.956i)11-s + (−5.66 + 3.22i)12-s + (−0.346 − 0.0929i)13-s + (0.562 + 4.37i)14-s + (−3.28 − 5.69i)15-s + (3.48 − 1.95i)16-s + (−1.10 + 1.91i)17-s + ⋯
L(s)  = 1  + (−0.127 − 0.991i)2-s + (1.81 − 0.487i)3-s + (−0.967 + 0.252i)4-s + (−0.233 − 0.870i)5-s + (−0.715 − 1.74i)6-s − 1.17·7-s + (0.374 + 0.927i)8-s + (2.20 − 1.27i)9-s + (−0.833 + 0.342i)10-s + (−0.288 − 0.288i)11-s + (−1.63 + 0.931i)12-s + (−0.0961 − 0.0257i)13-s + (0.150 + 1.16i)14-s + (−0.848 − 1.46i)15-s + (0.872 − 0.489i)16-s + (−0.267 + 0.463i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.589 + 0.807i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.589 + 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $-0.589 + 0.807i$
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: \((2,\ 304,\ (\ :1/2),\ -0.589 + 0.807i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.781593 - 1.53814i\)
\(L(\frac12)\) \(\approx\) \(0.781593 - 1.53814i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.180 + 1.40i)T \)
19 \( 1 + (-3.90 + 1.94i)T \)
good3 \( 1 + (-3.14 + 0.844i)T + (2.59 - 1.5i)T^{2} \)
5 \( 1 + (0.521 + 1.94i)T + (-4.33 + 2.5i)T^{2} \)
7 \( 1 + 3.12T + 7T^{2} \)
11 \( 1 + (0.956 + 0.956i)T + 11iT^{2} \)
13 \( 1 + (0.346 + 0.0929i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (1.10 - 1.91i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-3.84 - 6.66i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-7.14 - 1.91i)T + (25.1 + 14.5i)T^{2} \)
31 \( 1 - 0.247T + 31T^{2} \)
37 \( 1 + (-0.00564 - 0.00564i)T + 37iT^{2} \)
41 \( 1 + (5.41 - 9.38i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (6.12 - 1.64i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (8.47 - 4.89i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.47 + 12.9i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-1.49 - 5.59i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-0.282 + 1.05i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (0.0497 - 0.185i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (0.620 + 0.358i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-6.06 - 3.49i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.998 + 1.72i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (7.89 - 7.89i)T - 83iT^{2} \)
89 \( 1 + (7.84 + 13.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-10.0 - 5.80i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.61410389267652563742920087508, −9.999160282578972589371656977073, −9.496790677487352327024486736966, −8.648822182292800107663900801119, −8.063403910328356073103513825064, −6.84076426739811573601081141388, −4.84903013945263714602046226970, −3.47876640222157090398662289172, −2.90298209984039810530018745066, −1.28384960668402876690704936967, 2.79062708264191077256531161038, 3.59039970284235503458321755275, 4.84139981717538742818396452190, 6.68378304182920966405887228711, 7.22505279436122028521845054367, 8.279797792000626086631649039822, 9.068686713627591756143864431754, 9.944293343191501579521924157109, 10.42453159016046657980944536322, 12.47935613494553233130520926043

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