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Properties

Label 2-390-65.18-c1-0-7
Degree $2$
Conductor $390$
Sign $0.438 - 0.898i$
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 + (0.445 − 2.19i)5-s + (−0.707 + 0.707i)6-s − 0.115·7-s − i·8-s + 1.00i·9-s + (2.19 + 0.445i)10-s + (3.04 + 3.04i)11-s + (−0.707 − 0.707i)12-s + (2.56 + 2.53i)13-s − 0.115i·14-s + (1.86 − 1.23i)15-s + 16-s + (2.54 + 2.54i)17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.408 + 0.408i)3-s − 0.5·4-s + (0.199 − 0.979i)5-s + (−0.288 + 0.288i)6-s − 0.0438·7-s − 0.353i·8-s + 0.333i·9-s + (0.692 + 0.140i)10-s + (0.917 + 0.917i)11-s + (−0.204 − 0.204i)12-s + (0.711 + 0.702i)13-s − 0.0309i·14-s + (0.481 − 0.318i)15-s + 0.250·16-s + (0.617 + 0.617i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.438 - 0.898i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.438 - 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.438 - 0.898i$
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: \((2,\ 390,\ (\ :1/2),\ 0.438 - 0.898i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.36799 + 0.855043i\)
\(L(\frac12)\) \(\approx\) \(1.36799 + 0.855043i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.445 + 2.19i)T \)
13 \( 1 + (-2.56 - 2.53i)T \)
good7 \( 1 + 0.115T + 7T^{2} \)
11 \( 1 + (-3.04 - 3.04i)T + 11iT^{2} \)
17 \( 1 + (-2.54 - 2.54i)T + 17iT^{2} \)
19 \( 1 + (-2.06 - 2.06i)T + 19iT^{2} \)
23 \( 1 + (-2.14 + 2.14i)T - 23iT^{2} \)
29 \( 1 + 7.76iT - 29T^{2} \)
31 \( 1 + (-1 + i)T - 31iT^{2} \)
37 \( 1 + 10.7T + 37T^{2} \)
41 \( 1 + (-5.19 + 5.19i)T - 41iT^{2} \)
43 \( 1 + (3.16 - 3.16i)T - 43iT^{2} \)
47 \( 1 + 2.36T + 47T^{2} \)
53 \( 1 + (0.0480 + 0.0480i)T + 53iT^{2} \)
59 \( 1 + (-9.28 + 9.28i)T - 59iT^{2} \)
61 \( 1 + 8.36T + 61T^{2} \)
67 \( 1 - 12.2iT - 67T^{2} \)
71 \( 1 + (4.84 - 4.84i)T - 71iT^{2} \)
73 \( 1 + 0.721iT - 73T^{2} \)
79 \( 1 + 14.6iT - 79T^{2} \)
83 \( 1 + 2.49T + 83T^{2} \)
89 \( 1 + (4.24 - 4.24i)T - 89iT^{2} \)
97 \( 1 - 13.5iT - 97T^{2} \)
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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

−11.64157200243800898725542916667, −10.15953700571644689963811260649, −9.465837391762357423520225458587, −8.727321048500246992939921817858, −7.927657714417599434289399548726, −6.72495064429237990658070159421, −5.68988805370900946802369213082, −4.55752054782803503669852608073, −3.78400233380584513543623386497, −1.61834604418741823740797542860, 1.32098887135582128731039259608, 3.04485175487592781943673151321, 3.48890956583873731586745874032, 5.34567741044577527017257887309, 6.46462861855115976581457003338, 7.41425867516675226673348737117, 8.567808147414194733003575453227, 9.355262746751520305298464102530, 10.38978171515528529971712581246, 11.18231851640735134491585557435

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