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Properties

Label 2-390-65.47-c1-0-5
Degree $2$
Conductor $390$
Sign $0.698 - 0.715i$
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 + (1.95 + 1.09i)5-s + (0.707 + 0.707i)6-s + 0.720·7-s − i·8-s − 1.00i·9-s + (−1.09 + 1.95i)10-s + (1.34 − 1.34i)11-s + (−0.707 + 0.707i)12-s + (0.934 + 3.48i)13-s + 0.720i·14-s + (2.15 − 0.609i)15-s + 16-s + (0.906 − 0.906i)17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.408 − 0.408i)3-s − 0.5·4-s + (0.873 + 0.487i)5-s + (0.288 + 0.288i)6-s + 0.272·7-s − 0.353i·8-s − 0.333i·9-s + (−0.344 + 0.617i)10-s + (0.404 − 0.404i)11-s + (−0.204 + 0.204i)12-s + (0.259 + 0.965i)13-s + 0.192i·14-s + (0.555 − 0.157i)15-s + 0.250·16-s + (0.219 − 0.219i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.698 - 0.715i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.698 - 0.715i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.698 - 0.715i$
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: \((2,\ 390,\ (\ :1/2),\ 0.698 - 0.715i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.62009 + 0.682941i\)
\(L(\frac12)\) \(\approx\) \(1.62009 + 0.682941i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.95 - 1.09i)T \)
13 \( 1 + (-0.934 - 3.48i)T \)
good7 \( 1 - 0.720T + 7T^{2} \)
11 \( 1 + (-1.34 + 1.34i)T - 11iT^{2} \)
17 \( 1 + (-0.906 + 0.906i)T - 17iT^{2} \)
19 \( 1 + (-0.226 + 0.226i)T - 19iT^{2} \)
23 \( 1 + (2.23 + 2.23i)T + 23iT^{2} \)
29 \( 1 - 3.21iT - 29T^{2} \)
31 \( 1 + (-2.31 - 2.31i)T + 31iT^{2} \)
37 \( 1 - 4.44T + 37T^{2} \)
41 \( 1 + (6.15 + 6.15i)T + 41iT^{2} \)
43 \( 1 + (-0.523 - 0.523i)T + 43iT^{2} \)
47 \( 1 - 2.06T + 47T^{2} \)
53 \( 1 + (6.29 - 6.29i)T - 53iT^{2} \)
59 \( 1 + (7.63 + 7.63i)T + 59iT^{2} \)
61 \( 1 + 0.811T + 61T^{2} \)
67 \( 1 + 3.36iT - 67T^{2} \)
71 \( 1 + (11.5 + 11.5i)T + 71iT^{2} \)
73 \( 1 - 5.15iT - 73T^{2} \)
79 \( 1 + 12.6iT - 79T^{2} \)
83 \( 1 + 1.44T + 83T^{2} \)
89 \( 1 + (4.99 + 4.99i)T + 89iT^{2} \)
97 \( 1 + 11.7iT - 97T^{2} \)
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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

−11.43280735012885332940379002321, −10.36545415448910582870463050655, −9.335135940426279619682087828822, −8.702657159442272584799386693132, −7.58052278556825319063726200514, −6.63447334773858872254237340965, −6.00167841751568776030249942386, −4.68087128188041848177605853334, −3.23087143303340781226409903573, −1.69139037587554330461995021402, 1.47787749097977658336591849281, 2.80344420734706026954569766199, 4.13079944027343343428430559921, 5.16201123463645513770507219026, 6.16898974713020364931733133169, 7.83795529356417154854122933796, 8.616802665810398156492195281231, 9.684064289252390940461739463888, 10.04136820393413020554312215763, 11.10797307712473206923876505044

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