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Properties

Label 2-740-740.539-c0-0-1
Degree $2$
Conductor $740$
Sign $0.936 - 0.349i$
Analytic cond. $0.369308$
Root an. cond. $0.607707$
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  + (0.939 + 0.342i)2-s + (0.766 + 0.642i)4-s + (0.5 − 0.866i)5-s + (0.500 + 0.866i)8-s + (−0.766 + 0.642i)9-s + (0.766 − 0.642i)10-s + (−0.766 − 0.642i)13-s + (0.173 + 0.984i)16-s + (0.266 − 0.223i)17-s + (−0.939 + 0.342i)18-s + (0.939 − 0.342i)20-s + (−0.499 − 0.866i)25-s + (−0.5 − 0.866i)26-s + (−1.11 + 0.642i)29-s + (−0.173 + 0.984i)32-s + ⋯
L(s)  = 1  + (0.939 + 0.342i)2-s + (0.766 + 0.642i)4-s + (0.5 − 0.866i)5-s + (0.500 + 0.866i)8-s + (−0.766 + 0.642i)9-s + (0.766 − 0.642i)10-s + (−0.766 − 0.642i)13-s + (0.173 + 0.984i)16-s + (0.266 − 0.223i)17-s + (−0.939 + 0.342i)18-s + (0.939 − 0.342i)20-s + (−0.499 − 0.866i)25-s + (−0.5 − 0.866i)26-s + (−1.11 + 0.642i)29-s + (−0.173 + 0.984i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.936 - 0.349i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.936 - 0.349i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(740\)    =    \(2^{2} \cdot 5 \cdot 37\)
Sign: $0.936 - 0.349i$
Analytic conductor: \(0.369308\)
Root analytic conductor: \(0.607707\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{740} (539, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 740,\ (\ :0),\ 0.936 - 0.349i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.650251732\)
\(L(\frac12)\) \(\approx\) \(1.650251732\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.939 - 0.342i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (0.766 + 0.642i)T \)
good3 \( 1 + (0.766 - 0.642i)T^{2} \)
7 \( 1 + (-0.939 - 0.342i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)T^{2} \)
17 \( 1 + (-0.266 + 0.223i)T + (0.173 - 0.984i)T^{2} \)
19 \( 1 + (0.766 - 0.642i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (1.11 - 0.642i)T + (0.5 - 0.866i)T^{2} \)
31 \( 1 + T^{2} \)
41 \( 1 + (-0.266 - 0.223i)T + (0.173 + 0.984i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (1.70 - 0.300i)T + (0.939 - 0.342i)T^{2} \)
59 \( 1 + (-0.939 + 0.342i)T^{2} \)
61 \( 1 + (-0.826 + 0.984i)T + (-0.173 - 0.984i)T^{2} \)
67 \( 1 + (-0.939 - 0.342i)T^{2} \)
71 \( 1 + (-0.766 + 0.642i)T^{2} \)
73 \( 1 - 1.73iT - T^{2} \)
79 \( 1 + (-0.939 - 0.342i)T^{2} \)
83 \( 1 + (0.173 - 0.984i)T^{2} \)
89 \( 1 + (1.93 - 0.342i)T + (0.939 - 0.342i)T^{2} \)
97 \( 1 + (-0.766 + 1.32i)T + (-0.5 - 0.866i)T^{2} \)
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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

−10.82288565465767750410784162990, −9.781074187009740169457226542547, −8.730316232944507614144246552090, −7.951438594711571728767382330490, −7.11451624367898792105820143619, −5.78184061748954251746705590636, −5.36526090952423265772870701422, −4.49121342644326724142916923218, −3.13415677324725465830577282192, −2.01970820442044330070834099099, 1.97997818290919168181820299939, 2.99085623785292930398020984764, 3.91270979372496683527280602980, 5.20597500045679051619337218062, 6.04722431860522052048755942469, 6.75160217320674209692160968404, 7.64103226062183848042639987505, 9.122877021355103237672115317078, 9.863806560868431951503679524958, 10.66805983743670802790624428857

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