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Properties

Label 2-740-740.739-c0-0-5
Degree $2$
Conductor $740$
Sign $1$
Analytic cond. $0.369308$
Root an. cond. $0.607707$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 1.41·3-s + 4-s − 5-s + 1.41·6-s − 1.41·7-s + 8-s + 1.00·9-s − 10-s + 1.41·12-s − 1.41·14-s − 1.41·15-s + 16-s + 1.00·18-s − 1.41·19-s − 20-s − 2.00·21-s + 1.41·24-s + 25-s − 1.41·28-s − 1.41·30-s + 1.41·31-s + 32-s + 1.41·35-s + 1.00·36-s − 37-s − 1.41·38-s + ⋯
L(s)  = 1  + 2-s + 1.41·3-s + 4-s − 5-s + 1.41·6-s − 1.41·7-s + 8-s + 1.00·9-s − 10-s + 1.41·12-s − 1.41·14-s − 1.41·15-s + 16-s + 1.00·18-s − 1.41·19-s − 20-s − 2.00·21-s + 1.41·24-s + 25-s − 1.41·28-s − 1.41·30-s + 1.41·31-s + 32-s + 1.41·35-s + 1.00·36-s − 37-s − 1.41·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(740\)    =    \(2^{2} \cdot 5 \cdot 37\)
Sign: $1$
Analytic conductor: \(0.369308\)
Root analytic conductor: \(0.607707\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{740} (739, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 740,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.000388202\)
\(L(\frac12)\) \(\approx\) \(2.000388202\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−10.52885744978745449408111094146, −9.787106872475420971270468458364, −8.605350158533362248860228293391, −8.070643485654651624936043877967, −6.93284422066505501439304511836, −6.43518630521662583882435563706, −4.84990079529925671728874570259, −3.72439681459784654954251686624, −3.32676116978140480775099030689, −2.29026648523149598742622564635, 2.29026648523149598742622564635, 3.32676116978140480775099030689, 3.72439681459784654954251686624, 4.84990079529925671728874570259, 6.43518630521662583882435563706, 6.93284422066505501439304511836, 8.070643485654651624936043877967, 8.605350158533362248860228293391, 9.787106872475420971270468458364, 10.52885744978745449408111094146

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