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Properties

Label 2-390-65.8-c1-0-11
Degree $2$
Conductor $390$
Sign $0.220 + 0.975i$
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 + (−0.707 − 0.707i)3-s + 4-s + (−0.775 − 2.09i)5-s + (−0.707 − 0.707i)6-s + 0.946i·7-s + 8-s + 1.00i·9-s + (−0.775 − 2.09i)10-s + (3.13 − 3.13i)11-s + (−0.707 − 0.707i)12-s + (0.250 − 3.59i)13-s + 0.946i·14-s + (−0.934 + 2.03i)15-s + 16-s + (−3.42 − 3.42i)17-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.408 − 0.408i)3-s + 0.5·4-s + (−0.346 − 0.937i)5-s + (−0.288 − 0.288i)6-s + 0.357i·7-s + 0.353·8-s + 0.333i·9-s + (−0.245 − 0.663i)10-s + (0.944 − 0.944i)11-s + (−0.204 − 0.204i)12-s + (0.0696 − 0.997i)13-s + 0.253i·14-s + (−0.241 + 0.524i)15-s + 0.250·16-s + (−0.829 − 0.829i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.220 + 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.220 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.220 + 0.975i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ 0.220 + 0.975i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.33184 - 1.06440i\)
\(L(\frac12)\) \(\approx\) \(1.33184 - 1.06440i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 + (0.775 + 2.09i)T \)
13 \( 1 + (-0.250 + 3.59i)T \)
good7 \( 1 - 0.946iT - 7T^{2} \)
11 \( 1 + (-3.13 + 3.13i)T - 11iT^{2} \)
17 \( 1 + (3.42 + 3.42i)T + 17iT^{2} \)
19 \( 1 + (1.00 - 1.00i)T - 19iT^{2} \)
23 \( 1 + (-4.25 + 4.25i)T - 23iT^{2} \)
29 \( 1 - 5.39iT - 29T^{2} \)
31 \( 1 + (2.43 + 2.43i)T + 31iT^{2} \)
37 \( 1 - 11.9iT - 37T^{2} \)
41 \( 1 + (-6.03 - 6.03i)T + 41iT^{2} \)
43 \( 1 + (-0.242 + 0.242i)T - 43iT^{2} \)
47 \( 1 + 0.854iT - 47T^{2} \)
53 \( 1 + (-3.90 - 3.90i)T + 53iT^{2} \)
59 \( 1 + (-9.38 - 9.38i)T + 59iT^{2} \)
61 \( 1 - 9.79T + 61T^{2} \)
67 \( 1 + 3.51T + 67T^{2} \)
71 \( 1 + (6.99 + 6.99i)T + 71iT^{2} \)
73 \( 1 + 16.4T + 73T^{2} \)
79 \( 1 - 8.09iT - 79T^{2} \)
83 \( 1 + 15.2iT - 83T^{2} \)
89 \( 1 + (0.244 + 0.244i)T + 89iT^{2} \)
97 \( 1 + 2.67T + 97T^{2} \)
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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

−11.51279960908377671858480931473, −10.51615809345487982799427868805, −9.041529921634371596707595593646, −8.391934107761554277160358356350, −7.18137695722919767519372717454, −6.14505901799479890781363218032, −5.28011193888881372351819940521, −4.30518093931531348768752882314, −2.90887956184850015542584648489, −1.03325324063527052290837301259, 2.12662398711077638030483220386, 3.87080341953642488001882865652, 4.25128636964308948715105455295, 5.76183651879955161332485944666, 6.84442429696730452657862574644, 7.23305911377856792780893353762, 8.905966435864730478817706312986, 9.896627578595399477095163227149, 10.92549083700133305399104000770, 11.38813789975631861448970148296

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