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Properties

Label 2-390-15.2-c1-0-6
Degree $2$
Conductor $390$
Sign $0.662 - 0.749i$
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  + (0.707 − 0.707i)2-s + (1 + 1.41i)3-s − 1.00i·4-s + (−0.707 + 2.12i)5-s + (1.70 + 0.292i)6-s + (0.585 + 0.585i)7-s + (−0.707 − 0.707i)8-s + (−1.00 + 2.82i)9-s + (0.999 + 2i)10-s + 4i·11-s + (1.41 − 1.00i)12-s + (−0.707 + 0.707i)13-s + 0.828·14-s + (−3.70 + 1.12i)15-s − 1.00·16-s + (3 − 3i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (0.577 + 0.816i)3-s − 0.500i·4-s + (−0.316 + 0.948i)5-s + (0.696 + 0.119i)6-s + (0.221 + 0.221i)7-s + (−0.250 − 0.250i)8-s + (−0.333 + 0.942i)9-s + (0.316 + 0.632i)10-s + 1.20i·11-s + (0.408 − 0.288i)12-s + (−0.196 + 0.196i)13-s + 0.221·14-s + (−0.957 + 0.289i)15-s − 0.250·16-s + (0.727 − 0.727i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.662 - 0.749i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (287, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ 0.662 - 0.749i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.79959 + 0.811543i\)
\(L(\frac12)\) \(\approx\) \(1.79959 + 0.811543i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 0.707i)T \)
3 \( 1 + (-1 - 1.41i)T \)
5 \( 1 + (0.707 - 2.12i)T \)
13 \( 1 + (0.707 - 0.707i)T \)
good7 \( 1 + (-0.585 - 0.585i)T + 7iT^{2} \)
11 \( 1 - 4iT - 11T^{2} \)
17 \( 1 + (-3 + 3i)T - 17iT^{2} \)
19 \( 1 + 6.82iT - 19T^{2} \)
23 \( 1 + (-2.82 - 2.82i)T + 23iT^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 - 3.41T + 31T^{2} \)
37 \( 1 + (2.58 + 2.58i)T + 37iT^{2} \)
41 \( 1 - 2iT - 41T^{2} \)
43 \( 1 + (-1.24 + 1.24i)T - 43iT^{2} \)
47 \( 1 + (-2.24 + 2.24i)T - 47iT^{2} \)
53 \( 1 + (0.585 + 0.585i)T + 53iT^{2} \)
59 \( 1 + 4.48T + 59T^{2} \)
61 \( 1 - 4.82T + 61T^{2} \)
67 \( 1 + (0.828 + 0.828i)T + 67iT^{2} \)
71 \( 1 + 14.2iT - 71T^{2} \)
73 \( 1 + (11.0 - 11.0i)T - 73iT^{2} \)
79 \( 1 + 12.4iT - 79T^{2} \)
83 \( 1 + (-6.82 - 6.82i)T + 83iT^{2} \)
89 \( 1 + 11.1T + 89T^{2} \)
97 \( 1 + (-9.07 - 9.07i)T + 97iT^{2} \)
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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

−11.42324536784019387300646849255, −10.50234207983599253061433475622, −9.818950759281622376985219646440, −8.976869626556051753402839489271, −7.63164755779841768659563359090, −6.80001680515473308805609937200, −5.18758167396034018380986262370, −4.43692316008887322021167647127, −3.18615064066508699564036289858, −2.35247979678645319254139132156, 1.19324448058304920193204930480, 3.10079952322196621416810115887, 4.16038397790319089302222284634, 5.54116812885311822514635339121, 6.34672352747216141270714381340, 7.67869379192562561616960266002, 8.235919263959471435670853845723, 8.852330268786827653372102943239, 10.24424199466312455848909038959, 11.59982401360752979548692368062

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