Properties

Label 2-390-65.64-c1-0-3
Degree $2$
Conductor $390$
Sign $0.955 - 0.296i$
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  − 2-s i·3-s + 4-s + (1.48 + 1.67i)5-s + i·6-s + 4.15·7-s − 8-s − 9-s + (−1.48 − 1.67i)10-s + 5.35i·11-s i·12-s + (−3.28 − 1.48i)13-s − 4.15·14-s + (1.67 − 1.48i)15-s + 16-s + 1.19i·17-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577i·3-s + 0.5·4-s + (0.662 + 0.749i)5-s + 0.408i·6-s + 1.57·7-s − 0.353·8-s − 0.333·9-s + (−0.468 − 0.529i)10-s + 1.61i·11-s − 0.288i·12-s + (−0.911 − 0.410i)13-s − 1.11·14-s + (0.432 − 0.382i)15-s + 0.250·16-s + 0.289i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.955 - 0.296i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.955 - 0.296i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.955 - 0.296i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (259, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ 0.955 - 0.296i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.22078 + 0.184933i\)
\(L(\frac12)\) \(\approx\) \(1.22078 + 0.184933i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + iT \)
5 \( 1 + (-1.48 - 1.67i)T \)
13 \( 1 + (3.28 + 1.48i)T \)
good7 \( 1 - 4.15T + 7T^{2} \)
11 \( 1 - 5.35iT - 11T^{2} \)
17 \( 1 - 1.19iT - 17T^{2} \)
19 \( 1 - 2.80iT - 19T^{2} \)
23 \( 1 + 0.806iT - 23T^{2} \)
29 \( 1 - 7.50T + 29T^{2} \)
31 \( 1 + 7.92iT - 31T^{2} \)
37 \( 1 - 7.35T + 37T^{2} \)
41 \( 1 + 3.61iT - 41T^{2} \)
43 \( 1 + 6.57iT - 43T^{2} \)
47 \( 1 + 12.3T + 47T^{2} \)
53 \( 1 + 5.53iT - 53T^{2} \)
59 \( 1 - 12.8iT - 59T^{2} \)
61 \( 1 - 6.31T + 61T^{2} \)
67 \( 1 - 4.57T + 67T^{2} \)
71 \( 1 + 8.96iT - 71T^{2} \)
73 \( 1 + 4.08T + 73T^{2} \)
79 \( 1 + 12.4T + 79T^{2} \)
83 \( 1 + 10.0T + 83T^{2} \)
89 \( 1 + 5.03iT - 89T^{2} \)
97 \( 1 + 2.93T + 97T^{2} \)
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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

−11.31970716965439727820546573209, −10.26241891669683820820901472427, −9.770208548354188340301838279567, −8.390687166263841055388818422228, −7.60898621152526802463453672463, −6.97470460671354868868810866694, −5.74693926360488739577909309124, −4.58026749649490482774007913749, −2.47033827728668067269220462768, −1.70217467885167089520530718337, 1.19264159371482447054391284703, 2.74065682697962436023162922668, 4.62937998853158358638475562307, 5.27817941130839632766697663946, 6.50473579487026653508950134509, 8.029229563481685467851650386600, 8.525995908435874460791363936635, 9.359708604276950871385903994545, 10.27246335532001936136819351954, 11.28736543723534307442783276902

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