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Properties

Label 2-390-65.33-c1-0-1
Degree $2$
Conductor $390$
Sign $-0.988 - 0.150i$
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 + (−1.64 − 1.51i)5-s + (−0.258 + 0.965i)6-s + (−3.98 + 2.29i)7-s + 0.999·8-s + (0.866 − 0.499i)9-s + (2.13 − 0.672i)10-s + (0.765 + 2.85i)11-s + (−0.707 − 0.707i)12-s + (−3.40 + 1.17i)13-s − 4.59i·14-s + (−1.98 − 1.03i)15-s + (−0.5 + 0.866i)16-s + (−1.55 + 5.81i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.557 − 0.149i)3-s + (−0.249 − 0.433i)4-s + (−0.737 − 0.675i)5-s + (−0.105 + 0.394i)6-s + (−1.50 + 0.868i)7-s + 0.353·8-s + (0.288 − 0.166i)9-s + (0.674 − 0.212i)10-s + (0.230 + 0.860i)11-s + (−0.204 − 0.204i)12-s + (−0.945 + 0.327i)13-s − 1.22i·14-s + (−0.512 − 0.266i)15-s + (−0.125 + 0.216i)16-s + (−0.378 + 1.41i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.150i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.988 - 0.150i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $-0.988 - 0.150i$
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: \((2,\ 390,\ (\ :1/2),\ -0.988 - 0.150i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0263762 + 0.349301i\)
\(L(\frac12)\) \(\approx\) \(0.0263762 + 0.349301i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.64 + 1.51i)T \)
13 \( 1 + (3.40 - 1.17i)T \)
good7 \( 1 + (3.98 - 2.29i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.765 - 2.85i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (1.55 - 5.81i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (5.14 + 1.37i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (1.14 + 4.28i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (-8.57 - 4.94i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.23 + 2.23i)T + 31iT^{2} \)
37 \( 1 + (3.14 + 1.81i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.71 - 0.458i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (5.26 + 1.41i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + 6.65iT - 47T^{2} \)
53 \( 1 + (-0.101 - 0.101i)T + 53iT^{2} \)
59 \( 1 + (-0.305 + 1.14i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (3.63 + 6.29i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.53 - 11.3i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.20 + 4.51i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 - 15.1T + 73T^{2} \)
79 \( 1 - 6.34iT - 79T^{2} \)
83 \( 1 - 8.15iT - 83T^{2} \)
89 \( 1 + (-2.30 + 0.618i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-6.36 - 11.0i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.26713521851151742443464821900, −10.50339350596036495121509227398, −9.633062327976458995383784079857, −8.812226991747391252335081693594, −8.301793303187615694609521129272, −6.95673611394928366386134650380, −6.41553062961504690774579613352, −4.91249955564941149401037321873, −3.82928512300183704615139298306, −2.23579046134776791755043944601, 0.22849531376053958912477807284, 2.78235778650991264036934096775, 3.40738154096436505808996628789, 4.46060895618281821563369189858, 6.44269694117172498194721970541, 7.23225867764196304268569032880, 8.143690643914713162860739054499, 9.267089250786974628296682446413, 10.05445503486960648463179682672, 10.66067860953336425351949062845

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