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Properties

Label 2-390-39.5-c1-0-0
Degree $2$
Conductor $390$
Sign $-0.738 - 0.674i$
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 + (−1.58 + 0.707i)3-s + 1.00i·4-s + (−0.707 − 0.707i)5-s + (1.61 + 0.618i)6-s + (0.707 − 0.707i)8-s + (2.00 − 2.23i)9-s + 1.00i·10-s + (1.41 − 1.41i)11-s + (−0.707 − 1.58i)12-s + (0.418 + 3.58i)13-s + (1.61 + 0.618i)15-s − 1.00·16-s − 7.30·17-s + (−2.99 + 0.166i)18-s + (−5.16 + 5.16i)19-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (−0.912 + 0.408i)3-s + 0.500i·4-s + (−0.316 − 0.316i)5-s + (0.660 + 0.252i)6-s + (0.250 − 0.250i)8-s + (0.666 − 0.745i)9-s + 0.316i·10-s + (0.426 − 0.426i)11-s + (−0.204 − 0.456i)12-s + (0.116 + 0.993i)13-s + (0.417 + 0.159i)15-s − 0.250·16-s − 1.77·17-s + (−0.706 + 0.0393i)18-s + (−1.18 + 1.18i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.738 - 0.674i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.738 - 0.674i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $-0.738 - 0.674i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ -0.738 - 0.674i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0571069 + 0.147272i\)
\(L(\frac12)\) \(\approx\) \(0.0571069 + 0.147272i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.58 - 0.707i)T \)
5 \( 1 + (0.707 + 0.707i)T \)
13 \( 1 + (-0.418 - 3.58i)T \)
good7 \( 1 + 7iT^{2} \)
11 \( 1 + (-1.41 + 1.41i)T - 11iT^{2} \)
17 \( 1 + 7.30T + 17T^{2} \)
19 \( 1 + (5.16 - 5.16i)T - 19iT^{2} \)
23 \( 1 + 4.47T + 23T^{2} \)
29 \( 1 + 4.47iT - 29T^{2} \)
31 \( 1 + (3 - 3i)T - 31iT^{2} \)
37 \( 1 + (4 + 4i)T + 37iT^{2} \)
41 \( 1 + (-2.23 - 2.23i)T + 41iT^{2} \)
43 \( 1 - 7.16iT - 43T^{2} \)
47 \( 1 + (7.30 - 7.30i)T - 47iT^{2} \)
53 \( 1 - 1.41iT - 53T^{2} \)
59 \( 1 + (5.88 - 5.88i)T - 59iT^{2} \)
61 \( 1 - 10.6T + 61T^{2} \)
67 \( 1 + (-7.16 + 7.16i)T - 67iT^{2} \)
71 \( 1 + (-3.87 - 3.87i)T + 71iT^{2} \)
73 \( 1 + (5.16 + 5.16i)T + 73iT^{2} \)
79 \( 1 + 10T + 79T^{2} \)
83 \( 1 + (-5.65 - 5.65i)T + 83iT^{2} \)
89 \( 1 + (0.592 - 0.592i)T - 89iT^{2} \)
97 \( 1 + (1.16 - 1.16i)T - 97iT^{2} \)
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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

−11.41095687713238982829668626139, −10.96491905749480010028692554956, −9.915981090677610777242798700949, −9.059668914600270527530384009008, −8.237116444239579725139441951724, −6.81269338436042885439814493699, −6.05820037415172688065487419600, −4.48641444094425686499075859946, −3.88715925731325827312995461161, −1.81314899718030458089601765233, 0.13265617460980325265062695341, 2.11985201196596893417599245391, 4.22620033361907259745018017533, 5.29935748258280637832674056373, 6.53702416903514063514953488713, 6.93361180419447513672259264674, 8.074995037883201708109391584395, 8.998514229484911037507261386764, 10.27031139681549510500757589569, 10.90191209445210660100492773787

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