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Properties

Label 2-390-65.32-c1-0-12
Degree $2$
Conductor $390$
Sign $-0.464 + 0.885i$
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  + (0.5 − 0.866i)2-s + (−0.258 − 0.965i)3-s + (−0.499 − 0.866i)4-s + (1.50 − 1.64i)5-s + (−0.965 − 0.258i)6-s + (2.70 − 1.56i)7-s − 0.999·8-s + (−0.866 + 0.499i)9-s + (−0.674 − 2.13i)10-s + (0.628 − 0.168i)11-s + (−0.707 + 0.707i)12-s + (−1.78 + 3.13i)13-s − 3.12i·14-s + (−1.98 − 1.03i)15-s + (−0.5 + 0.866i)16-s + (−0.0844 − 0.0226i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.149 − 0.557i)3-s + (−0.249 − 0.433i)4-s + (0.674 − 0.737i)5-s + (−0.394 − 0.105i)6-s + (1.02 − 0.589i)7-s − 0.353·8-s + (−0.288 + 0.166i)9-s + (−0.213 − 0.674i)10-s + (0.189 − 0.0507i)11-s + (−0.204 + 0.204i)12-s + (−0.495 + 0.868i)13-s − 0.834i·14-s + (−0.512 − 0.266i)15-s + (−0.125 + 0.216i)16-s + (−0.0204 − 0.00548i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.464 + 0.885i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.464 + 0.885i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $-0.464 + 0.885i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ -0.464 + 0.885i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.911250 - 1.50670i\)
\(L(\frac12)\) \(\approx\) \(0.911250 - 1.50670i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (0.258 + 0.965i)T \)
5 \( 1 + (-1.50 + 1.64i)T \)
13 \( 1 + (1.78 - 3.13i)T \)
good7 \( 1 + (-2.70 + 1.56i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.628 + 0.168i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (0.0844 + 0.0226i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (0.264 - 0.986i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-2.28 + 0.611i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (7.96 + 4.59i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.40 - 1.40i)T - 31iT^{2} \)
37 \( 1 + (-6.58 - 3.80i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.901 + 3.36i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (2.60 - 9.73i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 - 4.74iT - 47T^{2} \)
53 \( 1 + (-6.10 + 6.10i)T - 53iT^{2} \)
59 \( 1 + (-12.7 - 3.42i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-3.40 - 5.89i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.42 - 9.39i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-13.0 - 3.48i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + 2.45T + 73T^{2} \)
79 \( 1 + 10.4iT - 79T^{2} \)
83 \( 1 + 1.51iT - 83T^{2} \)
89 \( 1 + (4.84 + 18.0i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-2.55 - 4.43i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.35088591041436467794699001276, −10.17758550548853571329451263482, −9.298896005982436314961246159366, −8.319231820985756571327621421436, −7.24255518133317387604698095867, −6.03198052846844560027686997912, −5.00484358810339014797609090814, −4.15673646950413166579085175416, −2.24175510572259094348932754427, −1.22034657004546872034158055556, 2.31348413534120937380356532112, 3.67105265435246911160078639283, 5.16626155258472896736905078836, 5.55929299993047037097199861950, 6.82441187806305476179649375773, 7.78966905044160206901524922309, 8.872111626214792444365956069114, 9.725347907922211248273427879553, 10.79798055121264065074247412477, 11.45046309149129984730374514329

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