Properties

Label 2-736-736.597-c0-0-1
Degree $2$
Conductor $736$
Sign $0.896 - 0.442i$
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  + (0.965 + 0.258i)2-s + (0.241 − 0.0999i)3-s + (0.866 + 0.499i)4-s + (0.258 − 0.0340i)6-s + (0.707 + 0.707i)8-s + (−0.658 + 0.658i)9-s + (0.258 + 0.0340i)12-s + (−0.607 − 1.46i)13-s + (0.500 + 0.866i)16-s + (−0.807 + 0.465i)18-s + (−0.707 + 0.707i)23-s + (0.241 + 0.0999i)24-s + (−0.707 − 0.707i)25-s + (−0.207 − 1.57i)26-s + (−0.192 + 0.465i)27-s + ⋯
L(s)  = 1  + (0.965 + 0.258i)2-s + (0.241 − 0.0999i)3-s + (0.866 + 0.499i)4-s + (0.258 − 0.0340i)6-s + (0.707 + 0.707i)8-s + (−0.658 + 0.658i)9-s + (0.258 + 0.0340i)12-s + (−0.607 − 1.46i)13-s + (0.500 + 0.866i)16-s + (−0.807 + 0.465i)18-s + (−0.707 + 0.707i)23-s + (0.241 + 0.0999i)24-s + (−0.707 − 0.707i)25-s + (−0.207 − 1.57i)26-s + (−0.192 + 0.465i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.896 - 0.442i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.896 - 0.442i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(736\)    =    \(2^{5} \cdot 23\)
Sign: $0.896 - 0.442i$
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: \((2,\ 736,\ (\ :0),\ 0.896 - 0.442i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.699534125\)
\(L(\frac12)\) \(\approx\) \(1.699534125\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 - 0.258i)T \)
23 \( 1 + (0.707 - 0.707i)T \)
good3 \( 1 + (-0.241 + 0.0999i)T + (0.707 - 0.707i)T^{2} \)
5 \( 1 + (0.707 + 0.707i)T^{2} \)
7 \( 1 - iT^{2} \)
11 \( 1 + (-0.707 - 0.707i)T^{2} \)
13 \( 1 + (0.607 + 1.46i)T + (-0.707 + 0.707i)T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + (0.707 - 0.707i)T^{2} \)
29 \( 1 + (-1.83 + 0.758i)T + (0.707 - 0.707i)T^{2} \)
31 \( 1 + 1.93T + T^{2} \)
37 \( 1 + (0.707 + 0.707i)T^{2} \)
41 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
43 \( 1 + (-0.707 - 0.707i)T^{2} \)
47 \( 1 + 1.73iT - T^{2} \)
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 \( 1 + (-0.366 - 0.366i)T + iT^{2} \)
73 \( 1 + (1.22 - 1.22i)T - iT^{2} \)
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

−10.71509901884131870542055293969, −10.05256214979725006139334356932, −8.620793294181934272067517705696, −7.86714050898267752185309290083, −7.25999462910086863648625826744, −5.89072752457882263119008129863, −5.43038824484416095240479759799, −4.27626659749624762890255795979, −3.10715049446684660757424627379, −2.22341647910113254941529772561, 1.88461695990926461549321251300, 3.05264592387260875150970868110, 4.05624904317598587457486643644, 4.94362710585338640583894882957, 6.09303723917043592777440729025, 6.74965871755292097678562178282, 7.78831089391763423714524351595, 9.036788639142083447476036296279, 9.667454526967782640252599129856, 10.74242641227842551698703839631

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