Properties

Label 2-390-65.2-c1-0-3
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: $\chi_{390} (67, \cdot )$
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

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

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