Properties

Label 2-736-736.597-c0-0-0
Degree $2$
Conductor $736$
Sign $-0.831 - 0.555i$
Analytic cond. $0.367311$
Root an. cond. $0.606062$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 + 0.707i)2-s + (−1.70 + 0.707i)3-s − 1.00i·4-s + (0.707 − 1.70i)6-s + (0.707 + 0.707i)8-s + (1.70 − 1.70i)9-s + (0.707 + 1.70i)12-s + (0.707 + 1.70i)13-s − 1.00·16-s + 2.41i·18-s + (−0.707 + 0.707i)23-s + (−1.70 − 0.707i)24-s + (−0.707 − 0.707i)25-s + (−1.70 − 0.707i)26-s + (−1 + 2.41i)27-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s + (−1.70 + 0.707i)3-s − 1.00i·4-s + (0.707 − 1.70i)6-s + (0.707 + 0.707i)8-s + (1.70 − 1.70i)9-s + (0.707 + 1.70i)12-s + (0.707 + 1.70i)13-s − 1.00·16-s + 2.41i·18-s + (−0.707 + 0.707i)23-s + (−1.70 − 0.707i)24-s + (−0.707 − 0.707i)25-s + (−1.70 − 0.707i)26-s + (−1 + 2.41i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.831 - 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.831 - 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(736\)    =    \(2^{5} \cdot 23\)
Sign: $-0.831 - 0.555i$
Analytic conductor: \(0.367311\)
Root analytic conductor: \(0.606062\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{736} (597, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 736,\ (\ :0),\ -0.831 - 0.555i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3238309256\)
\(L(\frac12)\) \(\approx\) \(0.3238309256\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
23 \( 1 + (0.707 - 0.707i)T \)
good3 \( 1 + (1.70 - 0.707i)T + (0.707 - 0.707i)T^{2} \)
5 \( 1 + (0.707 + 0.707i)T^{2} \)
7 \( 1 - iT^{2} \)
11 \( 1 + (-0.707 - 0.707i)T^{2} \)
13 \( 1 + (-0.707 - 1.70i)T + (-0.707 + 0.707i)T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + (0.707 - 0.707i)T^{2} \)
29 \( 1 + (0.707 - 0.292i)T + (0.707 - 0.707i)T^{2} \)
31 \( 1 - 1.41T + T^{2} \)
37 \( 1 + (0.707 + 0.707i)T^{2} \)
41 \( 1 + (1.41 - 1.41i)T - iT^{2} \)
43 \( 1 + (-0.707 - 0.707i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (-0.707 - 0.707i)T^{2} \)
59 \( 1 + (0.707 - 1.70i)T + (-0.707 - 0.707i)T^{2} \)
61 \( 1 + (-0.707 + 0.707i)T^{2} \)
67 \( 1 + (-0.707 + 0.707i)T^{2} \)
71 \( 1 + (-1 - i)T + iT^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + (0.707 - 0.707i)T^{2} \)
89 \( 1 - iT^{2} \)
97 \( 1 - T^{2} \)
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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

−10.87551440498601951503796082528, −9.962465863443904898479659821556, −9.508936688300050004498635108076, −8.452177939991638158849462132265, −7.19244605138376469750655553409, −6.34433345218821513984778243013, −5.91840444263563700200299094125, −4.78845074028024798547396599584, −4.09799367124061901634875571213, −1.47809779827531588874000999562, 0.58385159151448982742172486980, 1.94145929119225542962916292363, 3.58157605031551570664726485411, 4.95125689543422001088665067246, 5.88082297964113912247935691086, 6.73879356310672481849781192305, 7.73398050157421096783115849813, 8.334349125362362887914591266776, 9.811000594379173127103275559564, 10.48903078698393060901201986942

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