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Properties

Label 2-390-65.47-c1-0-10
Degree $2$
Conductor $390$
Sign $0.119 + 0.992i$
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 + (1.04 − 1.97i)5-s + (0.707 + 0.707i)6-s + 2.54·7-s + i·8-s − 1.00i·9-s + (−1.97 − 1.04i)10-s + (−1.89 + 1.89i)11-s + (0.707 − 0.707i)12-s + (1.52 − 3.26i)13-s − 2.54i·14-s + (0.656 + 2.13i)15-s + 16-s + (−1.27 + 1.27i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.408 + 0.408i)3-s − 0.5·4-s + (0.468 − 0.883i)5-s + (0.288 + 0.288i)6-s + 0.960·7-s + 0.353i·8-s − 0.333i·9-s + (−0.624 − 0.331i)10-s + (−0.571 + 0.571i)11-s + (0.204 − 0.204i)12-s + (0.424 − 0.905i)13-s − 0.679i·14-s + (0.169 + 0.551i)15-s + 0.250·16-s + (−0.310 + 0.310i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.119 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.119 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.119 + 0.992i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ 0.119 + 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.972346 - 0.862316i\)
\(L(\frac12)\) \(\approx\) \(0.972346 - 0.862316i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.04 + 1.97i)T \)
13 \( 1 + (-1.52 + 3.26i)T \)
good7 \( 1 - 2.54T + 7T^{2} \)
11 \( 1 + (1.89 - 1.89i)T - 11iT^{2} \)
17 \( 1 + (1.27 - 1.27i)T - 17iT^{2} \)
19 \( 1 + (-6.13 + 6.13i)T - 19iT^{2} \)
23 \( 1 + (0.312 + 0.312i)T + 23iT^{2} \)
29 \( 1 + 10.6iT - 29T^{2} \)
31 \( 1 + (-1 - i)T + 31iT^{2} \)
37 \( 1 + 3.18T + 37T^{2} \)
41 \( 1 + (-4.77 - 4.77i)T + 41iT^{2} \)
43 \( 1 + (-5.88 - 5.88i)T + 43iT^{2} \)
47 \( 1 + 5.18T + 47T^{2} \)
53 \( 1 + (6.13 - 6.13i)T - 53iT^{2} \)
59 \( 1 + (-7.78 - 7.78i)T + 59iT^{2} \)
61 \( 1 - 6.41T + 61T^{2} \)
67 \( 1 - 11.3iT - 67T^{2} \)
71 \( 1 + (7.30 + 7.30i)T + 71iT^{2} \)
73 \( 1 - 14.5iT - 73T^{2} \)
79 \( 1 + 3.04iT - 79T^{2} \)
83 \( 1 + 3.42T + 83T^{2} \)
89 \( 1 + (2.37 + 2.37i)T + 89iT^{2} \)
97 \( 1 + 7.40iT - 97T^{2} \)
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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

−11.22244718042315188125575969751, −10.18747786950673748279042635488, −9.518529342387573238480574362355, −8.507140759490000716562662085935, −7.63933610868652656109398579912, −5.88369597337753656920215140268, −5.02252572104878990333061494960, −4.34518312368736172763694444044, −2.60718869040013214402876617418, −1.03515030369852661744986314256, 1.71082687026627518456743258674, 3.47311333546592841735033770078, 5.08314315948737832658941844200, 5.80981130212861648157112275556, 6.85092006949591642582257149123, 7.59718870615311498279888001359, 8.538972461144040660915758252323, 9.681176745410641919024261895587, 10.76618672771881267624516805420, 11.36563416213686796781647656647

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