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Properties

Label 2-390-65.9-c1-0-14
Degree $2$
Conductor $390$
Sign $0.974 - 0.222i$
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 + (0.866 + 0.5i)3-s + (0.499 + 0.866i)4-s + (−1.5 − 1.65i)5-s + (0.499 + 0.866i)6-s + (3.73 − 2.15i)7-s + 0.999i·8-s + (0.499 + 0.866i)9-s + (−0.469 − 2.18i)10-s + (3.15 − 5.47i)11-s + 0.999i·12-s + (−0.866 + 3.5i)13-s + 4.31·14-s + (−0.469 − 2.18i)15-s + (−0.5 + 0.866i)16-s + (−6.61 + 3.81i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.499 + 0.288i)3-s + (0.249 + 0.433i)4-s + (−0.670 − 0.741i)5-s + (0.204 + 0.353i)6-s + (1.41 − 0.815i)7-s + 0.353i·8-s + (0.166 + 0.288i)9-s + (−0.148 − 0.691i)10-s + (0.952 − 1.64i)11-s + 0.288i·12-s + (−0.240 + 0.970i)13-s + 1.15·14-s + (−0.121 − 0.564i)15-s + (−0.125 + 0.216i)16-s + (−1.60 + 0.925i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 - 0.222i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.974 - 0.222i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.974 - 0.222i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ 0.974 - 0.222i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.25623 + 0.254376i\)
\(L(\frac12)\) \(\approx\) \(2.25623 + 0.254376i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.866 - 0.5i)T \)
5 \( 1 + (1.5 + 1.65i)T \)
13 \( 1 + (0.866 - 3.5i)T \)
good7 \( 1 + (-3.73 + 2.15i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-3.15 + 5.47i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (6.61 - 3.81i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.15 - 3.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.45 - 0.841i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.5 + 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + (6.61 + 3.81i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.341 + 0.591i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.00 - 1.15i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 1.68iT - 47T^{2} \)
53 \( 1 - 4.68iT - 53T^{2} \)
59 \( 1 + (2.31 + 4.01i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.341 + 0.591i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.01 - 3.47i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (4.15 + 7.20i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 0.683iT - 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 - 8.31iT - 83T^{2} \)
89 \( 1 + (5.63 - 9.75i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.19 - 3i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.31757640538207231377258607438, −10.86881754996230615573154145228, −9.091322409391666113783954531451, −8.494283224017642191399261681486, −7.76938998940483419448690887061, −6.64325857218809747479024657541, −5.27628700686129031084739951942, −4.18691900298102542978135652686, −3.78592254697986734201503904384, −1.60333887787027055187157742871, 1.91566676168322572870801759898, 2.88444488410576779205934733830, 4.38525972764236098810736960851, 5.08395689205893323863532423052, 6.80853667063063481342394487344, 7.32227371890678088133009189521, 8.506111972719694338208746032037, 9.423476984191963689270263719164, 10.66552701782420193211414596195, 11.53688003490097191608655694244

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