Properties

Label 2-390-65.7-c1-0-6
Degree $2$
Conductor $390$
Sign $0.525 - 0.850i$
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 + (0.258 + 0.965i)3-s + (0.499 + 0.866i)4-s + (1.17 − 1.90i)5-s + (−0.258 + 0.965i)6-s + (1.97 + 3.41i)7-s + 0.999i·8-s + (−0.866 + 0.499i)9-s + (1.96 − 1.06i)10-s + (−1.48 − 5.56i)11-s + (−0.707 + 0.707i)12-s + (0.730 + 3.53i)13-s + 3.94i·14-s + (2.14 + 0.637i)15-s + (−0.5 + 0.866i)16-s + (3.85 + 1.03i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.149 + 0.557i)3-s + (0.249 + 0.433i)4-s + (0.523 − 0.851i)5-s + (−0.105 + 0.394i)6-s + (0.745 + 1.29i)7-s + 0.353i·8-s + (−0.288 + 0.166i)9-s + (0.621 − 0.336i)10-s + (−0.449 − 1.67i)11-s + (−0.204 + 0.204i)12-s + (0.202 + 0.979i)13-s + 1.05i·14-s + (0.553 + 0.164i)15-s + (−0.125 + 0.216i)16-s + (0.936 + 0.250i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.525 - 0.850i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ 0.525 - 0.850i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.94344 + 1.08419i\)
\(L(\frac12)\) \(\approx\) \(1.94344 + 1.08419i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.258 - 0.965i)T \)
5 \( 1 + (-1.17 + 1.90i)T \)
13 \( 1 + (-0.730 - 3.53i)T \)
good7 \( 1 + (-1.97 - 3.41i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.48 + 5.56i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (-3.85 - 1.03i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (2.08 + 0.558i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (4.18 - 1.12i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (-5.04 - 2.91i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.83 + 2.83i)T + 31iT^{2} \)
37 \( 1 + (1.82 - 3.16i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.0577 - 0.0154i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-2.76 + 10.3i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + 13.4T + 47T^{2} \)
53 \( 1 + (-4.15 + 4.15i)T - 53iT^{2} \)
59 \( 1 + (-0.509 + 1.89i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (1.42 + 2.47i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.35 + 1.36i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3.31 + 12.3i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + 4.59iT - 73T^{2} \)
79 \( 1 - 2.49iT - 79T^{2} \)
83 \( 1 - 12.0T + 83T^{2} \)
89 \( 1 + (8.00 - 2.14i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-8.26 + 4.76i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.67956796682066051777785972183, −10.67753946158799259015789737740, −9.394871423616443413495804267126, −8.517098717709533836481278996798, −8.179344246988656383274327991868, −6.23594271116808041835373887181, −5.55235827169119234455021638056, −4.83628829609406952031614747578, −3.48701843591205211040111524355, −2.03719159209019725168882966550, 1.54242107241138421945686553428, 2.76714302399659932802651562808, 4.11157990133571249910755025196, 5.24684814975580750753244613980, 6.46936114706162841730930677246, 7.39130484225460368484362538959, 7.963955002578464848642968381062, 9.892955276190054842852738405785, 10.28010852051582735717995906940, 11.10733836671051279602875510864

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