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Properties

Label 2-780-13.4-c1-0-7
Degree $2$
Conductor $780$
Sign $-0.00641 + 0.999i$
Analytic cond. $6.22833$
Root an. cond. $2.49566$
Motivic weight $1$
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)3-s − i·5-s + (−2.18 + 1.25i)7-s + (−0.499 − 0.866i)9-s + (0.590 + 0.341i)11-s + (−3.08 − 1.86i)13-s + (0.866 + 0.5i)15-s + (−2.39 − 4.15i)17-s + (5.65 − 3.26i)19-s − 2.51i·21-s + (3.68 − 6.38i)23-s − 25-s + 0.999·27-s + (−2.31 + 4.00i)29-s + 0.600i·31-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s − 0.447i·5-s + (−0.824 + 0.475i)7-s + (−0.166 − 0.288i)9-s + (0.178 + 0.102i)11-s + (−0.856 − 0.516i)13-s + (0.223 + 0.129i)15-s + (−0.581 − 1.00i)17-s + (1.29 − 0.748i)19-s − 0.549i·21-s + (0.768 − 1.33i)23-s − 0.200·25-s + 0.192·27-s + (−0.429 + 0.744i)29-s + 0.107i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 780 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00641 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 780 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00641 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(780\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 13\)
Sign: $-0.00641 + 0.999i$
Analytic conductor: \(6.22833\)
Root analytic conductor: \(2.49566\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{780} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 780,\ (\ :1/2),\ -0.00641 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.540771 - 0.544249i\)
\(L(\frac12)\) \(\approx\) \(0.540771 - 0.544249i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 + iT \)
13 \( 1 + (3.08 + 1.86i)T \)
good7 \( 1 + (2.18 - 1.25i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.590 - 0.341i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (2.39 + 4.15i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-5.65 + 3.26i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.68 + 6.38i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.31 - 4.00i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 0.600iT - 31T^{2} \)
37 \( 1 + (8.85 + 5.11i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.31 + 0.758i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.47 + 6.01i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 4.73iT - 47T^{2} \)
53 \( 1 - 10.3T + 53T^{2} \)
59 \( 1 + (-8.19 + 4.73i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.66 + 9.80i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.81 + 2.20i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.291 + 0.168i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 4.50iT - 73T^{2} \)
79 \( 1 + 9.19T + 79T^{2} \)
83 \( 1 - 10.0iT - 83T^{2} \)
89 \( 1 + (-11.2 - 6.51i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.97 + 4.60i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.998154081421848729799330801741, −9.212964164087928492524484637059, −8.759156294398344214248223454453, −7.30540314648672309895938893822, −6.66603549461512455429100664975, −5.29717562944458899343452415546, −4.96882951450913543394369452463, −3.52517707516495961564036255334, −2.52340532411149461527086331210, −0.40010761981483010194213582068, 1.53823765138649913682329658082, 3.00677533638737660281070678991, 3.98329991401844379823774898082, 5.33972553745805284477006268946, 6.26592194252501752253272763770, 7.07795896495830808509084566953, 7.64029807273659204141828596234, 8.873352940987226584957176906869, 9.843787880192748917485811678908, 10.36846533396042528396653361049

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