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Properties

Label 2-390-65.28-c1-0-13
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: $\chi_{390} (223, \cdot )$
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

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

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