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Properties

Label 2-390-65.37-c1-0-0
Degree $2$
Conductor $390$
Sign $-0.204 - 0.978i$
Analytic cond. $3.11416$
Root an. cond. $1.76469$
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  + (−0.866 + 0.5i)2-s + (−0.965 − 0.258i)3-s + (0.499 − 0.866i)4-s + (2.12 + 0.707i)5-s + (0.965 − 0.258i)6-s + (−1.56 + 2.71i)7-s + 0.999i·8-s + (0.866 + 0.499i)9-s + (−2.19 + 0.448i)10-s + (−3.99 − 1.06i)11-s + (−0.707 + 0.707i)12-s + (3.53 − 0.707i)13-s − 3.13i·14-s + (−1.86 − 1.23i)15-s + (−0.5 − 0.866i)16-s + (1.66 + 6.22i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−0.557 − 0.149i)3-s + (0.249 − 0.433i)4-s + (0.948 + 0.316i)5-s + (0.394 − 0.105i)6-s + (−0.591 + 1.02i)7-s + 0.353i·8-s + (0.288 + 0.166i)9-s + (−0.692 + 0.141i)10-s + (−1.20 − 0.322i)11-s + (−0.204 + 0.204i)12-s + (0.980 − 0.196i)13-s − 0.836i·14-s + (−0.481 − 0.318i)15-s + (−0.125 − 0.216i)16-s + (0.404 + 1.51i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.204 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.204 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $-0.204 - 0.978i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ -0.204 - 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.494802 + 0.608907i\)
\(L(\frac12)\) \(\approx\) \(0.494802 + 0.608907i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 + (0.965 + 0.258i)T \)
5 \( 1 + (-2.12 - 0.707i)T \)
13 \( 1 + (-3.53 + 0.707i)T \)
good7 \( 1 + (1.56 - 2.71i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (3.99 + 1.06i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-1.66 - 6.22i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (0.560 + 2.09i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (1.49 - 5.57i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (7.51 - 4.33i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.54 - 4.54i)T + 31iT^{2} \)
37 \( 1 + (-0.351 - 0.609i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.133 - 0.497i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (3.38 - 0.905i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 - 4.69T + 47T^{2} \)
53 \( 1 + (8.92 - 8.92i)T - 53iT^{2} \)
59 \( 1 + (-9.64 + 2.58i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-4.20 + 7.28i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.13 + 4.12i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-6.47 + 1.73i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + 11.5iT - 73T^{2} \)
79 \( 1 - 10.6iT - 79T^{2} \)
83 \( 1 + 13.1T + 83T^{2} \)
89 \( 1 + (-4.02 + 15.0i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (3.60 + 2.08i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.22778135910005626572008628154, −10.59741357301787682877252675438, −9.777794761076147573381940975736, −8.849361570367360828400470208709, −7.944873449732042118582683148279, −6.64745495352594858784001460473, −5.83508161365641391326765945417, −5.40648630342726445712571100611, −3.17547184407504068945141532815, −1.74687074324416179597910832543, 0.67958252510744377457143937226, 2.42246699272583060347722261145, 3.98016911780624111293283810285, 5.27319309496861308107571865781, 6.34186185968782647586929359109, 7.30024013124282137743566203611, 8.386630228987725573407598499658, 9.681816317482392442057518038380, 10.02176499126619820132752963677, 10.82343544604051152601979550225

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