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Properties

Label 2-390-39.11-c1-0-15
Degree $2$
Conductor $390$
Sign $0.999 + 0.00620i$
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  + (0.965 + 0.258i)2-s + (1.30 − 1.13i)3-s + (0.866 + 0.499i)4-s + (0.707 − 0.707i)5-s + (1.55 − 0.762i)6-s + (1.23 + 4.62i)7-s + (0.707 + 0.707i)8-s + (0.404 − 2.97i)9-s + (0.866 − 0.500i)10-s + (−0.551 + 2.05i)11-s + (1.69 − 0.334i)12-s + (−2.60 − 2.49i)13-s + 4.78i·14-s + (0.117 − 1.72i)15-s + (0.500 + 0.866i)16-s + (1.43 − 2.48i)17-s + ⋯
L(s)  = 1  + (0.683 + 0.183i)2-s + (0.753 − 0.657i)3-s + (0.433 + 0.249i)4-s + (0.316 − 0.316i)5-s + (0.634 − 0.311i)6-s + (0.468 + 1.74i)7-s + (0.249 + 0.249i)8-s + (0.134 − 0.990i)9-s + (0.273 − 0.158i)10-s + (−0.166 + 0.620i)11-s + (0.490 − 0.0964i)12-s + (−0.723 − 0.690i)13-s + 1.27i·14-s + (0.0302 − 0.446i)15-s + (0.125 + 0.216i)16-s + (0.347 − 0.602i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00620i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00620i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.999 + 0.00620i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ 0.999 + 0.00620i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.61922 - 0.00812265i\)
\(L(\frac12)\) \(\approx\) \(2.61922 - 0.00812265i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 - 0.258i)T \)
3 \( 1 + (-1.30 + 1.13i)T \)
5 \( 1 + (-0.707 + 0.707i)T \)
13 \( 1 + (2.60 + 2.49i)T \)
good7 \( 1 + (-1.23 - 4.62i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (0.551 - 2.05i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (-1.43 + 2.48i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (5.39 - 1.44i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (2.61 + 4.53i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-5.71 + 3.29i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.28 + 1.28i)T + 31iT^{2} \)
37 \( 1 + (7.85 + 2.10i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-0.581 - 0.155i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-1.82 - 1.05i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.63 - 5.63i)T + 47iT^{2} \)
53 \( 1 - 5.77iT - 53T^{2} \)
59 \( 1 + (4.37 - 1.17i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-2.74 + 4.75i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.57 - 13.3i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (0.537 + 2.00i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (3.31 - 3.31i)T - 73iT^{2} \)
79 \( 1 - 16.3T + 79T^{2} \)
83 \( 1 + (-4.14 + 4.14i)T - 83iT^{2} \)
89 \( 1 + (-2.23 + 8.35i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (5.63 - 1.50i)T + (84.0 - 48.5i)T^{2} \)
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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

−11.91262691534330140844397112796, −10.35258821150848088846906991862, −9.226888671478071758607130838055, −8.427477173092342775767825680840, −7.67665554081710522731066157330, −6.41467430037318405353291461091, −5.55796540315195361862626214590, −4.49994691299854016853136067928, −2.72864301203980315017886537730, −2.10876735330061079585030382715, 1.88436181335149022954560263925, 3.39739212426584856665139107613, 4.19131673835310241653935906340, 5.11987541015577157589108836182, 6.64232512083795636223542407503, 7.51387503743385901876272861660, 8.518849818005372492699055165813, 9.795045412479181990472325035245, 10.58516762459550482779586088441, 10.93187152285967616807058380555

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