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Properties

Label 2-740-185.154-c0-0-0
Degree $2$
Conductor $740$
Sign $0.763 - 0.646i$
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  + 3-s + i·5-s + i·7-s − i·11-s + i·15-s + i·21-s − 25-s − 27-s + (1 − i)31-s − i·33-s − 35-s + i·37-s − i·41-s + (1 − i)43-s − i·47-s + ⋯
L(s)  = 1  + 3-s + i·5-s + i·7-s − i·11-s + i·15-s + i·21-s − 25-s − 27-s + (1 − i)31-s − i·33-s − 35-s + i·37-s − i·41-s + (1 − i)43-s − i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.763 - 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.763 - 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(740\)    =    \(2^{2} \cdot 5 \cdot 37\)
Sign: $0.763 - 0.646i$
Analytic conductor: \(0.369308\)
Root analytic conductor: \(0.607707\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{740} (709, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 740,\ (\ :0),\ 0.763 - 0.646i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.246664927\)
\(L(\frac12)\) \(\approx\) \(1.246664927\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−10.63325544955693430019050879680, −9.709738251650673909448266769049, −8.815956066759734611264887578989, −8.287091089627404246905712027391, −7.34791937619071667550532329305, −6.22516514386988278338421210863, −5.53143500709999724914375613130, −3.87989686433611595479549379133, −2.94998505337625106834770821887, −2.28419346217152213566924936899, 1.46806011096337313505749075438, 2.85769584937893753582032870896, 4.11579110887827576871678713366, 4.74968310816742776401947003273, 6.07874611895756882370790525401, 7.37184725362495879523472201970, 7.888967216271389877968668404181, 8.831829086715782214702007489982, 9.480101897896462060185475216704, 10.25568745673041232392408024981

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