Properties

Label 2-390-65.7-c1-0-1
Degree $2$
Conductor $390$
Sign $-0.442 - 0.896i$
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  + (0.866 + 0.5i)2-s + (0.258 + 0.965i)3-s + (0.499 + 0.866i)4-s + (−0.860 + 2.06i)5-s + (−0.258 + 0.965i)6-s + (0.259 + 0.450i)7-s + 0.999i·8-s + (−0.866 + 0.499i)9-s + (−1.77 + 1.35i)10-s + (0.222 + 0.830i)11-s + (−0.707 + 0.707i)12-s + (−3.14 − 1.76i)13-s + 0.519i·14-s + (−2.21 − 0.296i)15-s + (−0.5 + 0.866i)16-s + (4.08 + 1.09i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.149 + 0.557i)3-s + (0.249 + 0.433i)4-s + (−0.384 + 0.923i)5-s + (−0.105 + 0.394i)6-s + (0.0982 + 0.170i)7-s + 0.353i·8-s + (−0.288 + 0.166i)9-s + (−0.561 + 0.429i)10-s + (0.0670 + 0.250i)11-s + (−0.204 + 0.204i)12-s + (−0.872 − 0.489i)13-s + 0.138i·14-s + (−0.572 − 0.0766i)15-s + (−0.125 + 0.216i)16-s + (0.990 + 0.265i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.442 - 0.896i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.442 - 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $-0.442 - 0.896i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ -0.442 - 0.896i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.945876 + 1.52218i\)
\(L(\frac12)\) \(\approx\) \(0.945876 + 1.52218i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 + (-0.258 - 0.965i)T \)
5 \( 1 + (0.860 - 2.06i)T \)
13 \( 1 + (3.14 + 1.76i)T \)
good7 \( 1 + (-0.259 - 0.450i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.222 - 0.830i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (-4.08 - 1.09i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (0.0538 + 0.0144i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-0.177 + 0.0474i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (-6.27 - 3.62i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.09 - 3.09i)T + 31iT^{2} \)
37 \( 1 + (-0.112 + 0.195i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.0280 - 0.00751i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-1.24 + 4.63i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 - 5.53T + 47T^{2} \)
53 \( 1 + (-2.49 + 2.49i)T - 53iT^{2} \)
59 \( 1 + (-1.31 + 4.89i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-7.21 - 12.5i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (11.8 + 6.85i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.78 - 6.64i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + 2.40iT - 73T^{2} \)
79 \( 1 + 16.0iT - 79T^{2} \)
83 \( 1 + 4.83T + 83T^{2} \)
89 \( 1 + (-14.9 + 3.99i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (5.45 - 3.14i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.82451864271890396543460683051, −10.54400904679394646778242706530, −10.09982694190053196591735064801, −8.711014942462950965038497437289, −7.71517728555022947678085262613, −6.92399506435564544550001683776, −5.74605192176425557750103653203, −4.73119135709565972582583351811, −3.56285376673025630353101908338, −2.62001858020081417702737803680, 1.03746128871270151968926580658, 2.62106265471268997003509163513, 4.05425286940499104606651857960, 5.00494150106440324947210712186, 6.08435009031708084783507311490, 7.31934482760910469861535925545, 8.126070114049493383249732940336, 9.231744202066045550603715610880, 10.12413777031959871299630632283, 11.42795069188445794928133438535

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