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Properties

Label 2-740-740.327-c0-0-0
Degree $2$
Conductor $740$
Sign $0.988 + 0.148i$
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  − i·2-s − 4-s + i·5-s + i·8-s + i·9-s + 10-s + 16-s + 2·17-s + 18-s − i·20-s − 25-s + (−1 + i)29-s − i·32-s − 2i·34-s − i·36-s − i·37-s + ⋯
L(s)  = 1  − i·2-s − 4-s + i·5-s + i·8-s + i·9-s + 10-s + 16-s + 2·17-s + 18-s − i·20-s − 25-s + (−1 + i)29-s − i·32-s − 2i·34-s − i·36-s − i·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.148i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.148i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(740\)    =    \(2^{2} \cdot 5 \cdot 37\)
Sign: $0.988 + 0.148i$
Analytic conductor: \(0.369308\)
Root analytic conductor: \(0.607707\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{740} (327, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 740,\ (\ :0),\ 0.988 + 0.148i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8687453087\)
\(L(\frac12)\) \(\approx\) \(0.8687453087\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−10.57813614425359603213208867668, −10.02904491616834568720827439624, −9.107829446704716072264366223918, −7.909115558792915112305275725088, −7.39903171173445046694591710730, −5.87958463340048995073013933301, −5.09421290498138495675666327786, −3.76033614175963782059800560877, −2.94503992374843256784392617654, −1.77624255501839965468835099483, 1.08203045151597496778684276714, 3.43889508708290576948611783833, 4.34697812104293897324953036548, 5.48481998210826103065977876402, 5.99987968102262097613565408915, 7.21895728688323744159414885855, 7.996143520107648956585897053323, 8.788542336512110385131804257079, 9.584726181061094742404565105427, 10.12558537338091247494720843241

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