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Properties

Label 2-390-65.33-c1-0-8
Degree $2$
Conductor $390$
Sign $0.716 + 0.697i$
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  + (0.5 − 0.866i)2-s + (0.965 − 0.258i)3-s + (−0.499 − 0.866i)4-s + (−0.0421 + 2.23i)5-s + (0.258 − 0.965i)6-s + (2.21 − 1.27i)7-s − 0.999·8-s + (0.866 − 0.499i)9-s + (1.91 + 1.15i)10-s + (0.240 + 0.897i)11-s + (−0.707 − 0.707i)12-s + (3.05 − 1.91i)13-s − 2.55i·14-s + (0.537 + 2.17i)15-s + (−0.5 + 0.866i)16-s + (0.495 − 1.84i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.557 − 0.149i)3-s + (−0.249 − 0.433i)4-s + (−0.0188 + 0.999i)5-s + (0.105 − 0.394i)6-s + (0.836 − 0.482i)7-s − 0.353·8-s + (0.288 − 0.166i)9-s + (0.605 + 0.365i)10-s + (0.0725 + 0.270i)11-s + (−0.204 − 0.204i)12-s + (0.846 − 0.531i)13-s − 0.682i·14-s + (0.138 + 0.560i)15-s + (−0.125 + 0.216i)16-s + (0.120 − 0.448i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.716 + 0.697i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.716 + 0.697i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.716 + 0.697i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ 0.716 + 0.697i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.90067 - 0.771726i\)
\(L(\frac12)\) \(\approx\) \(1.90067 - 0.771726i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (-0.965 + 0.258i)T \)
5 \( 1 + (0.0421 - 2.23i)T \)
13 \( 1 + (-3.05 + 1.91i)T \)
good7 \( 1 + (-2.21 + 1.27i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.240 - 0.897i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (-0.495 + 1.84i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-0.321 - 0.0861i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-0.516 - 1.92i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (-1.58 - 0.914i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.423 + 0.423i)T + 31iT^{2} \)
37 \( 1 + (6.15 + 3.55i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (10.7 - 2.88i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (1.38 + 0.371i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 - 7.62iT - 47T^{2} \)
53 \( 1 + (0.567 + 0.567i)T + 53iT^{2} \)
59 \( 1 + (-1.37 + 5.13i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (5.39 + 9.34i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.90 - 11.9i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.55 - 13.2i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 - 16.1T + 73T^{2} \)
79 \( 1 + 10.6iT - 79T^{2} \)
83 \( 1 - 6.12iT - 83T^{2} \)
89 \( 1 + (-6.90 + 1.85i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-1.12 - 1.94i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.12958823503459518015166851460, −10.49643810082172586520854459768, −9.595692123146066909228501192751, −8.387982735128239904750484006590, −7.50724760989011900794152314380, −6.51303182518641438679241339873, −5.18101943612647775256695161138, −3.89397034035204707959000158464, −2.98276332564841662842557725520, −1.59649887871542475554794683446, 1.73403590585548094611021206087, 3.56871661181340014438200716509, 4.64186430269176918271803124529, 5.47044129453414461374350788369, 6.64373851936446305361071411917, 8.008114719516992174539029076540, 8.551709829195321177824021389846, 9.143318383782099234469598966784, 10.46564383388899287784330419256, 11.71136907220048970198575008680

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