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Properties

Label 2-390-65.28-c1-0-8
Degree $2$
Conductor $390$
Sign $0.931 + 0.364i$
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 + (2.23 + 0.0421i)5-s + (0.258 + 0.965i)6-s + (1.27 − 2.21i)7-s − 0.999i·8-s + (−0.866 − 0.499i)9-s + (1.95 − 1.08i)10-s + (0.240 − 0.897i)11-s + (0.707 + 0.707i)12-s + (−1.91 + 3.05i)13-s − 2.55i·14-s + (−0.619 + 2.14i)15-s + (−0.5 − 0.866i)16-s + (1.84 − 0.495i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (−0.149 + 0.557i)3-s + (0.249 − 0.433i)4-s + (0.999 + 0.0188i)5-s + (0.105 + 0.394i)6-s + (0.482 − 0.836i)7-s − 0.353i·8-s + (−0.288 − 0.166i)9-s + (0.618 − 0.341i)10-s + (0.0725 − 0.270i)11-s + (0.204 + 0.204i)12-s + (−0.531 + 0.846i)13-s − 0.682i·14-s + (−0.159 + 0.554i)15-s + (−0.125 − 0.216i)16-s + (0.448 − 0.120i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.931 + 0.364i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.931 + 0.364i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.931 + 0.364i$
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: \((2,\ 390,\ (\ :1/2),\ 0.931 + 0.364i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.11572 - 0.399772i\)
\(L(\frac12)\) \(\approx\) \(2.11572 - 0.399772i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.23 - 0.0421i)T \)
13 \( 1 + (1.91 - 3.05i)T \)
good7 \( 1 + (-1.27 + 2.21i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.240 + 0.897i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (-1.84 + 0.495i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (0.321 - 0.0861i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-1.92 - 0.516i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (1.58 - 0.914i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.423 - 0.423i)T - 31iT^{2} \)
37 \( 1 + (-3.55 - 6.15i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (10.7 + 2.88i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (0.371 + 1.38i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + 7.62T + 47T^{2} \)
53 \( 1 + (0.567 + 0.567i)T + 53iT^{2} \)
59 \( 1 + (1.37 + 5.13i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (5.39 - 9.34i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (11.9 - 6.90i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.55 + 13.2i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 - 16.1iT - 73T^{2} \)
79 \( 1 + 10.6iT - 79T^{2} \)
83 \( 1 - 6.12T + 83T^{2} \)
89 \( 1 + (6.90 + 1.85i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (1.94 + 1.12i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.23040492801174172027344507486, −10.36115277440289767257920161215, −9.763809431517847694577507535883, −8.756942511034954530347605073209, −7.28082500165344772467924415194, −6.29612974148617930975262303043, −5.19592293062937711954407402634, −4.42943422188450226245446436963, −3.12535482737682934650289398135, −1.58718228081539179189964308758, 1.85925344111451321956572188594, 2.98959653680451799933911188613, 4.88899903235955990435676665696, 5.57929873320522886637568300223, 6.41138278565218264140202187831, 7.51188751210853516294149427018, 8.439414058111132375076763382144, 9.486685199572178702678925161247, 10.54771386958002892098947782989, 11.62667235588559253033409071990

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