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Properties

Label 2-390-65.57-c1-0-12
Degree $2$
Conductor $390$
Sign $0.432 + 0.901i$
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 + (−2.22 − 0.227i)5-s + (0.707 − 0.707i)6-s − 4.05i·7-s + 8-s − 1.00i·9-s + (−2.22 − 0.227i)10-s + (1.63 + 1.63i)11-s + (0.707 − 0.707i)12-s + (3.18 − 1.68i)13-s − 4.05i·14-s + (−1.73 + 1.41i)15-s + 16-s + (0.932 − 0.932i)17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.408 − 0.408i)3-s + 0.5·4-s + (−0.994 − 0.101i)5-s + (0.288 − 0.288i)6-s − 1.53i·7-s + 0.353·8-s − 0.333i·9-s + (−0.703 − 0.0720i)10-s + (0.493 + 0.493i)11-s + (0.204 − 0.204i)12-s + (0.883 − 0.467i)13-s − 1.08i·14-s + (−0.447 + 0.364i)15-s + 0.250·16-s + (0.226 − 0.226i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.432 + 0.901i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.432 + 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.432 + 0.901i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ 0.432 + 0.901i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.73060 - 1.08975i\)
\(L(\frac12)\) \(\approx\) \(1.73060 - 1.08975i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.22 + 0.227i)T \)
13 \( 1 + (-3.18 + 1.68i)T \)
good7 \( 1 + 4.05iT - 7T^{2} \)
11 \( 1 + (-1.63 - 1.63i)T + 11iT^{2} \)
17 \( 1 + (-0.932 + 0.932i)T - 17iT^{2} \)
19 \( 1 + (4.66 + 4.66i)T + 19iT^{2} \)
23 \( 1 + (-3.93 - 3.93i)T + 23iT^{2} \)
29 \( 1 - 6.86iT - 29T^{2} \)
31 \( 1 + (6.67 - 6.67i)T - 31iT^{2} \)
37 \( 1 - 8.04iT - 37T^{2} \)
41 \( 1 + (-6.69 + 6.69i)T - 41iT^{2} \)
43 \( 1 + (2.58 + 2.58i)T + 43iT^{2} \)
47 \( 1 + 0.559iT - 47T^{2} \)
53 \( 1 + (6.34 - 6.34i)T - 53iT^{2} \)
59 \( 1 + (2.29 - 2.29i)T - 59iT^{2} \)
61 \( 1 - 6.48T + 61T^{2} \)
67 \( 1 - 7.13T + 67T^{2} \)
71 \( 1 + (-5.56 + 5.56i)T - 71iT^{2} \)
73 \( 1 - 7.36T + 73T^{2} \)
79 \( 1 - 1.91iT - 79T^{2} \)
83 \( 1 - 0.718iT - 83T^{2} \)
89 \( 1 + (4.02 - 4.02i)T - 89iT^{2} \)
97 \( 1 + 6.60T + 97T^{2} \)
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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

−11.01362235292128966576443052617, −10.75455628144459914092524819447, −9.182678261598029920972511146379, −8.153674918799452195810440808892, −7.10781350005803909051523303898, −6.82791624727879644427735451616, −5.02278392585028426515790151770, −3.97870938715301883597993808993, −3.26806712172980784583627242423, −1.19268503725986286167511322839, 2.26931558250495156595000518431, 3.55202264316065212124950670226, 4.31293856440506382695455931706, 5.70033026805611793384239646676, 6.46643075331482265219633454549, 7.991423733316624257553985626461, 8.580824804638590656474358774741, 9.516114428068631790445758782133, 11.03391154864915614117351781890, 11.38661909916693822662630838199

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