Properties

Label 2-736-736.413-c0-0-1
Degree $2$
Conductor $736$
Sign $-0.0654 - 0.997i$
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  + (−0.258 + 0.965i)2-s + (1.46 + 0.607i)3-s + (−0.866 − 0.499i)4-s + (−0.965 + 1.25i)6-s + (0.707 − 0.707i)8-s + (1.07 + 1.07i)9-s + (−0.965 − 1.25i)12-s + (−0.0999 + 0.241i)13-s + (0.500 + 0.866i)16-s + (−1.31 + 0.758i)18-s + (−0.707 − 0.707i)23-s + (1.46 − 0.607i)24-s + (−0.707 + 0.707i)25-s + (−0.207 − 0.158i)26-s + (0.314 + 0.758i)27-s + ⋯
L(s)  = 1  + (−0.258 + 0.965i)2-s + (1.46 + 0.607i)3-s + (−0.866 − 0.499i)4-s + (−0.965 + 1.25i)6-s + (0.707 − 0.707i)8-s + (1.07 + 1.07i)9-s + (−0.965 − 1.25i)12-s + (−0.0999 + 0.241i)13-s + (0.500 + 0.866i)16-s + (−1.31 + 0.758i)18-s + (−0.707 − 0.707i)23-s + (1.46 − 0.607i)24-s + (−0.707 + 0.707i)25-s + (−0.207 − 0.158i)26-s + (0.314 + 0.758i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0654 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0654 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(736\)    =    \(2^{5} \cdot 23\)
Sign: $-0.0654 - 0.997i$
Analytic conductor: \(0.367311\)
Root analytic conductor: \(0.606062\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{736} (413, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 736,\ (\ :0),\ -0.0654 - 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.207896303\)
\(L(\frac12)\) \(\approx\) \(1.207896303\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 - 0.965i)T \)
23 \( 1 + (0.707 + 0.707i)T \)
good3 \( 1 + (-1.46 - 0.607i)T + (0.707 + 0.707i)T^{2} \)
5 \( 1 + (0.707 - 0.707i)T^{2} \)
7 \( 1 + iT^{2} \)
11 \( 1 + (-0.707 + 0.707i)T^{2} \)
13 \( 1 + (0.0999 - 0.241i)T + (-0.707 - 0.707i)T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + (0.707 + 0.707i)T^{2} \)
29 \( 1 + (1.12 + 0.465i)T + (0.707 + 0.707i)T^{2} \)
31 \( 1 - 0.517T + T^{2} \)
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 \( 1 + (1.36 - 1.36i)T - iT^{2} \)
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

−10.23949542281191560049678548423, −9.700966932887271237595405823650, −8.998431629146286004604933107959, −8.232607873325309976251222697421, −7.64804986893239938450750006299, −6.61872401950830653747181370996, −5.44827130654023069441803285799, −4.32478509269783461279542829414, −3.58527587416105969649875528592, −2.09647978089545162019235842218, 1.58143335091342308629469492507, 2.56904947319582517151505818612, 3.46199461403233600106200137027, 4.41832488652380825119173026029, 5.93766914238446500906426370389, 7.54419335367687225109725074836, 7.78854684569036509903663907765, 8.849730078359333897558565197612, 9.361346446187031353613313932025, 10.21166414736004496123189234051

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