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Properties

Label 2-390-65.63-c1-0-5
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: $\chi_{390} (193, \cdot )$
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

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

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