Properties

Label 2-390-65.33-c1-0-9
Degree $2$
Conductor $390$
Sign $0.141 + 0.990i$
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 + (−0.965 + 0.258i)3-s + (−0.499 − 0.866i)4-s + (2.06 − 0.860i)5-s + (−0.258 + 0.965i)6-s + (0.450 − 0.259i)7-s − 0.999·8-s + (0.866 − 0.499i)9-s + (0.286 − 2.21i)10-s + (0.222 + 0.830i)11-s + (0.707 + 0.707i)12-s + (1.76 − 3.14i)13-s − 0.519i·14-s + (−1.77 + 1.36i)15-s + (−0.5 + 0.866i)16-s + (1.09 − 4.08i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.557 + 0.149i)3-s + (−0.249 − 0.433i)4-s + (0.923 − 0.384i)5-s + (−0.105 + 0.394i)6-s + (0.170 − 0.0982i)7-s − 0.353·8-s + (0.288 − 0.166i)9-s + (0.0907 − 0.701i)10-s + (0.0670 + 0.250i)11-s + (0.204 + 0.204i)12-s + (0.489 − 0.872i)13-s − 0.138i·14-s + (−0.457 + 0.352i)15-s + (−0.125 + 0.216i)16-s + (0.265 − 0.990i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.141 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.141 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.141 + 0.990i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ 0.141 + 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.17854 - 1.02254i\)
\(L(\frac12)\) \(\approx\) \(1.17854 - 1.02254i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.965 - 0.258i)T \)
5 \( 1 + (-2.06 + 0.860i)T \)
13 \( 1 + (-1.76 + 3.14i)T \)
good7 \( 1 + (-0.450 + 0.259i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.222 - 0.830i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (-1.09 + 4.08i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-0.0538 - 0.0144i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-0.0474 - 0.177i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (6.27 + 3.62i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.09 - 3.09i)T + 31iT^{2} \)
37 \( 1 + (0.195 + 0.112i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.0280 - 0.00751i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-4.63 - 1.24i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + 5.53iT - 47T^{2} \)
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 \( 1 + (6.85 - 11.8i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.78 - 6.64i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 - 2.40T + 73T^{2} \)
79 \( 1 - 16.0iT - 79T^{2} \)
83 \( 1 + 4.83iT - 83T^{2} \)
89 \( 1 + (14.9 - 3.99i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-3.14 - 5.45i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.12669801884829851676634881260, −10.23406452737164926298465510843, −9.615500694764850690116772244241, −8.592501829306654589890473185046, −7.19880736327610205778767427212, −5.90076073284372683030220953660, −5.30895945125242870055392633428, −4.20532389497615709283885852428, −2.68850053366666629642925900731, −1.12858712204585361847388118189, 1.84485513729291646860286915492, 3.59956435332031577737970042338, 4.93132349427972033094046379570, 5.97522672704525881239446584502, 6.47477078112159185564264989867, 7.57039755198106831747653792734, 8.737518414152405352335838336373, 9.651640913535912295537959561116, 10.73594011280189687579723234177, 11.47468105970830833131256466892

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