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Properties

Label 2-390-65.18-c1-0-0
Degree $2$
Conductor $390$
Sign $-0.588 + 0.808i$
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  + i·2-s + (0.707 + 0.707i)3-s − 4-s + (−2.23 + 0.0440i)5-s + (−0.707 + 0.707i)6-s − 4.35·7-s − i·8-s + 1.00i·9-s + (−0.0440 − 2.23i)10-s + (−0.747 − 0.747i)11-s + (−0.707 − 0.707i)12-s + (3.32 − 1.38i)13-s − 4.35i·14-s + (−1.61 − 1.54i)15-s + 16-s + (−4.23 − 4.23i)17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.408 + 0.408i)3-s − 0.5·4-s + (−0.999 + 0.0196i)5-s + (−0.288 + 0.288i)6-s − 1.64·7-s − 0.353i·8-s + 0.333i·9-s + (−0.0139 − 0.706i)10-s + (−0.225 − 0.225i)11-s + (−0.204 − 0.204i)12-s + (0.922 − 0.385i)13-s − 1.16i·14-s + (−0.416 − 0.400i)15-s + 0.250·16-s + (−1.02 − 1.02i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.588 + 0.808i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.588 + 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $-0.588 + 0.808i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (343, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ -0.588 + 0.808i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0488716 - 0.0960811i\)
\(L(\frac12)\) \(\approx\) \(0.0488716 - 0.0960811i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 + (2.23 - 0.0440i)T \)
13 \( 1 + (-3.32 + 1.38i)T \)
good7 \( 1 + 4.35T + 7T^{2} \)
11 \( 1 + (0.747 + 0.747i)T + 11iT^{2} \)
17 \( 1 + (4.23 + 4.23i)T + 17iT^{2} \)
19 \( 1 + (2.33 + 2.33i)T + 19iT^{2} \)
23 \( 1 + (5.77 - 5.77i)T - 23iT^{2} \)
29 \( 1 - 8.52iT - 29T^{2} \)
31 \( 1 + (-1 + i)T - 31iT^{2} \)
37 \( 1 + 7.52T + 37T^{2} \)
41 \( 1 + (5.52 - 5.52i)T - 41iT^{2} \)
43 \( 1 + (2.48 - 2.48i)T - 43iT^{2} \)
47 \( 1 + 2.03T + 47T^{2} \)
53 \( 1 + (1.80 + 1.80i)T + 53iT^{2} \)
59 \( 1 + (-2.51 + 2.51i)T - 59iT^{2} \)
61 \( 1 - 15.0T + 61T^{2} \)
67 \( 1 + 4.15iT - 67T^{2} \)
71 \( 1 + (8.14 - 8.14i)T - 71iT^{2} \)
73 \( 1 + 0.693iT - 73T^{2} \)
79 \( 1 + 6.33iT - 79T^{2} \)
83 \( 1 - 0.717T + 83T^{2} \)
89 \( 1 + (-10.7 + 10.7i)T - 89iT^{2} \)
97 \( 1 - 11.0iT - 97T^{2} \)
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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

−11.92220704632440127645962139150, −10.87947598191608970370975538320, −9.891506523790700141882880636638, −8.988539350623535604621435639369, −8.307307086806129864789287759406, −7.13689633124678592825880691229, −6.44179681808081714636057179633, −5.11715104744608085117398813605, −3.81621917178467426408086381154, −3.12681006156015015056474028638, 0.06452049434120595888920203796, 2.23369168910741702763953607931, 3.60576156695004547069260553314, 4.14243583587055525275042763530, 6.14331924868872118569570421584, 6.85303654193918268275160390398, 8.284211177795923995390792299586, 8.729664730273208234259925051793, 9.981650317736698202521482004342, 10.64648355809663102704104029654

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