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Properties

Label 2-740-740.103-c0-0-0
Degree $2$
Conductor $740$
Sign $-0.501 - 0.864i$
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  + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (0.5 + 0.866i)5-s − 0.999·8-s + (−0.866 + 0.5i)9-s + (−0.499 + 0.866i)10-s + (−0.5 − 0.866i)16-s + (0.866 − 0.5i)17-s + (−0.866 − 0.499i)18-s − 0.999·20-s + (−0.499 + 0.866i)25-s + (0.366 − 0.366i)29-s + (0.499 − 0.866i)32-s + (0.866 + 0.499i)34-s − 0.999i·36-s + (0.5 − 0.866i)37-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (0.5 + 0.866i)5-s − 0.999·8-s + (−0.866 + 0.5i)9-s + (−0.499 + 0.866i)10-s + (−0.5 − 0.866i)16-s + (0.866 − 0.5i)17-s + (−0.866 − 0.499i)18-s − 0.999·20-s + (−0.499 + 0.866i)25-s + (0.366 − 0.366i)29-s + (0.499 − 0.866i)32-s + (0.866 + 0.499i)34-s − 0.999i·36-s + (0.5 − 0.866i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.501 - 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.501 - 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(740\)    =    \(2^{2} \cdot 5 \cdot 37\)
Sign: $-0.501 - 0.864i$
Analytic conductor: \(0.369308\)
Root analytic conductor: \(0.607707\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{740} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 740,\ (\ :0),\ -0.501 - 0.864i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.159264263\)
\(L(\frac12)\) \(\approx\) \(1.159264263\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
good3 \( 1 + (0.866 - 0.5i)T^{2} \)
7 \( 1 + (0.866 - 0.5i)T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.866 + 0.5i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (-0.366 + 0.366i)T - iT^{2} \)
31 \( 1 - iT^{2} \)
41 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + (-0.366 + 1.36i)T + (-0.866 - 0.5i)T^{2} \)
59 \( 1 + (0.866 + 0.5i)T^{2} \)
61 \( 1 + (1.86 - 0.5i)T + (0.866 - 0.5i)T^{2} \)
67 \( 1 + (-0.866 + 0.5i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 + (-1 + i)T - iT^{2} \)
79 \( 1 + (-0.866 + 0.5i)T^{2} \)
83 \( 1 + (0.866 + 0.5i)T^{2} \)
89 \( 1 + (-0.5 - 0.133i)T + (0.866 + 0.5i)T^{2} \)
97 \( 1 + 1.73iT - T^{2} \)
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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

−10.97064404882227763332563421188, −9.872801313763093227550469840766, −9.085977957989118253671985713452, −7.976684975844693786211637533087, −7.40781059875336606638964225870, −6.29347289060867962412606591786, −5.73357111130998480011409002232, −4.74767562791917414200360054618, −3.38705646808784961804865351769, −2.53150957590109695689596552981, 1.18650984517537811091964955515, 2.59197125658397918543742063870, 3.71429227809838839125525544229, 4.81289940774018761250187815729, 5.68424962340427503687408290502, 6.32466421380189644502150852783, 7.985002539397403184741002760297, 8.890435757547006821265797883768, 9.487191724931224059379683108775, 10.34680491369231432876513830953

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