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Properties

Label 2-390-65.37-c1-0-11
Degree $2$
Conductor $390$
Sign $0.469 + 0.882i$
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 + (−0.900 − 2.04i)5-s + (0.965 − 0.258i)6-s + (0.0298 − 0.0517i)7-s − 0.999i·8-s + (0.866 + 0.499i)9-s + (−1.80 − 1.32i)10-s + (2.65 + 0.710i)11-s + (0.707 − 0.707i)12-s + (0.914 − 3.48i)13-s − 0.0597i·14-s + (−0.340 − 2.20i)15-s + (−0.5 − 0.866i)16-s + (1.22 + 4.56i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.557 + 0.149i)3-s + (0.249 − 0.433i)4-s + (−0.402 − 0.915i)5-s + (0.394 − 0.105i)6-s + (0.0112 − 0.0195i)7-s − 0.353i·8-s + (0.288 + 0.166i)9-s + (−0.570 − 0.418i)10-s + (0.799 + 0.214i)11-s + (0.204 − 0.204i)12-s + (0.253 − 0.967i)13-s − 0.0159i·14-s + (−0.0879 − 0.570i)15-s + (−0.125 − 0.216i)16-s + (0.296 + 1.10i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.469 + 0.882i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.469 + 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.469 + 0.882i$
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: \((2,\ 390,\ (\ :1/2),\ 0.469 + 0.882i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.86824 - 1.12250i\)
\(L(\frac12)\) \(\approx\) \(1.86824 - 1.12250i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.900 + 2.04i)T \)
13 \( 1 + (-0.914 + 3.48i)T \)
good7 \( 1 + (-0.0298 + 0.0517i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.65 - 0.710i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-1.22 - 4.56i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (2.19 + 8.18i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (0.837 - 3.12i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (3.06 - 1.76i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-5.45 - 5.45i)T + 31iT^{2} \)
37 \( 1 + (-0.224 - 0.388i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.07 - 7.76i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (3.94 - 1.05i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 - 8.10T + 47T^{2} \)
53 \( 1 + (3.62 - 3.62i)T - 53iT^{2} \)
59 \( 1 + (8.24 - 2.20i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-5.07 + 8.79i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (8.49 - 4.90i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.76 - 0.474i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 - 11.0iT - 73T^{2} \)
79 \( 1 + 12.6iT - 79T^{2} \)
83 \( 1 - 0.810T + 83T^{2} \)
89 \( 1 + (-3.49 + 13.0i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-9.22 - 5.32i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.27143696504421214068180656575, −10.33244652266248692596583767757, −9.244454600305688579856720258216, −8.541336173281863798370639541961, −7.49937519968706475534560021481, −6.22851745060648532182000424146, −4.99324509976030971583433561986, −4.14302896005339121372530059577, −3.06476254341107077742315349842, −1.36151386340530283500969306962, 2.18446263386740363327990973183, 3.54531645266995313837682336631, 4.24226469636214063188450779961, 5.94024946423812547841379928289, 6.73468701737299888545130329908, 7.59536036215833313058533896185, 8.497857946571902508436190222615, 9.607683607960149567020379399573, 10.64967039114933407371930072553, 11.82437315890645650533091117638

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