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Properties

Label 2-304-76.75-c1-0-8
Degree $2$
Conductor $304$
Sign $i$
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  − 5-s − 4.35i·7-s − 3·9-s − 4.35i·11-s + 7·17-s + 4.35i·19-s − 8.71i·23-s − 4·25-s + 4.35i·35-s + 13.0i·43-s + 3·45-s − 4.35i·47-s − 12.0·49-s + 4.35i·55-s + 15·61-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.64i·7-s − 9-s − 1.31i·11-s + 1.69·17-s + 0.999i·19-s − 1.81i·23-s − 0.800·25-s + 0.736i·35-s + 1.99i·43-s + 0.447·45-s − 0.635i·47-s − 1.71·49-s + 0.587i·55-s + 1.92·61-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $i$
Analytic conductor: \(2.42745\)
Root analytic conductor: \(1.55802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (303, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 304,\ (\ :1/2),\ i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.709639 - 0.709639i\)
\(L(\frac12)\) \(\approx\) \(0.709639 - 0.709639i\)
\(L(\frac{3}{2})\) 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 \( 1 - 4.35iT \)
good3 \( 1 + 3T^{2} \)
5 \( 1 + T + 5T^{2} \)
7 \( 1 + 4.35iT - 7T^{2} \)
11 \( 1 + 4.35iT - 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 - 7T + 17T^{2} \)
23 \( 1 + 8.71iT - 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 - 13.0iT - 43T^{2} \)
47 \( 1 + 4.35iT - 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 15T + 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 11T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 8.71iT - 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−11.36424100874035080602044629917, −10.64039820529012766146601931497, −9.810156314211485736748144542945, −8.246007557642040916600789169367, −7.935279911670733220867378369606, −6.57397978870308106344754277290, −5.54429051267460212401686320132, −4.04036788852722839709519649446, −3.18985105245041001800734306256, −0.75067994626145005201323780568, 2.20554819837438602305096567011, 3.45263935766864084060798214387, 5.17400013551212808950091505439, 5.76173891115666305871457994825, 7.25110734562622309623475704069, 8.176959562550778912654720271459, 9.169588155260978087686422243889, 9.865224886415717308099031813263, 11.37058711974801084999360835271, 11.93844245095129171959488032957

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