Properties

Label 2-690-115.49-c1-0-6
Degree $2$
Conductor $690$
Sign $0.561 - 0.827i$
Analytic cond. $5.50967$
Root an. cond. $2.34727$
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  + (0.989 − 0.142i)2-s + (−0.909 − 0.415i)3-s + (0.959 − 0.281i)4-s + (1.08 + 1.95i)5-s + (−0.959 − 0.281i)6-s + (0.378 + 0.588i)7-s + (0.909 − 0.415i)8-s + (0.654 + 0.755i)9-s + (1.34 + 1.78i)10-s + (−0.745 + 5.18i)11-s + (−0.989 − 0.142i)12-s + (−1.92 + 2.99i)13-s + (0.457 + 0.528i)14-s + (−0.171 − 2.22i)15-s + (0.841 − 0.540i)16-s + (0.765 − 2.60i)17-s + ⋯
L(s)  = 1  + (0.699 − 0.100i)2-s + (−0.525 − 0.239i)3-s + (0.479 − 0.140i)4-s + (0.483 + 0.875i)5-s + (−0.391 − 0.115i)6-s + (0.142 + 0.222i)7-s + (0.321 − 0.146i)8-s + (0.218 + 0.251i)9-s + (0.426 + 0.563i)10-s + (−0.224 + 1.56i)11-s + (−0.285 − 0.0410i)12-s + (−0.534 + 0.832i)13-s + (0.122 + 0.141i)14-s + (−0.0441 − 0.575i)15-s + (0.210 − 0.135i)16-s + (0.185 − 0.632i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.561 - 0.827i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.561 - 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.561 - 0.827i$
Analytic conductor: \(5.50967\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1/2),\ 0.561 - 0.827i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.75072 + 0.928303i\)
\(L(\frac12)\) \(\approx\) \(1.75072 + 0.928303i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.989 + 0.142i)T \)
3 \( 1 + (0.909 + 0.415i)T \)
5 \( 1 + (-1.08 - 1.95i)T \)
23 \( 1 + (-0.308 + 4.78i)T \)
good7 \( 1 + (-0.378 - 0.588i)T + (-2.90 + 6.36i)T^{2} \)
11 \( 1 + (0.745 - 5.18i)T + (-10.5 - 3.09i)T^{2} \)
13 \( 1 + (1.92 - 2.99i)T + (-5.40 - 11.8i)T^{2} \)
17 \( 1 + (-0.765 + 2.60i)T + (-14.3 - 9.19i)T^{2} \)
19 \( 1 + (3.73 - 1.09i)T + (15.9 - 10.2i)T^{2} \)
29 \( 1 + (-8.22 - 2.41i)T + (24.3 + 15.6i)T^{2} \)
31 \( 1 + (-2.66 - 5.82i)T + (-20.3 + 23.4i)T^{2} \)
37 \( 1 + (-4.34 + 3.76i)T + (5.26 - 36.6i)T^{2} \)
41 \( 1 + (1.31 - 1.51i)T + (-5.83 - 40.5i)T^{2} \)
43 \( 1 + (-4.36 - 1.99i)T + (28.1 + 32.4i)T^{2} \)
47 \( 1 + 5.47iT - 47T^{2} \)
53 \( 1 + (-1.05 - 1.64i)T + (-22.0 + 48.2i)T^{2} \)
59 \( 1 + (9.17 + 5.89i)T + (24.5 + 53.6i)T^{2} \)
61 \( 1 + (-5.71 - 12.5i)T + (-39.9 + 46.1i)T^{2} \)
67 \( 1 + (3.44 - 0.495i)T + (64.2 - 18.8i)T^{2} \)
71 \( 1 + (0.0479 + 0.333i)T + (-68.1 + 20.0i)T^{2} \)
73 \( 1 + (3.65 + 12.4i)T + (-61.4 + 39.4i)T^{2} \)
79 \( 1 + (3.98 + 2.55i)T + (32.8 + 71.8i)T^{2} \)
83 \( 1 + (-12.2 + 10.6i)T + (11.8 - 82.1i)T^{2} \)
89 \( 1 + (-3.75 + 8.22i)T + (-58.2 - 67.2i)T^{2} \)
97 \( 1 + (-5.51 - 4.78i)T + (13.8 + 96.0i)T^{2} \)
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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

−10.44333044534471439811333650702, −10.22248397296266459505253958038, −9.027214770879701535485830204155, −7.56628166468155421720379273651, −6.84345722711947631074524616840, −6.27887477778143476691720436702, −5.02572646228644686511383330542, −4.39451906472271662890850782940, −2.73498205794778990825085452868, −1.91599802663257541247173877565, 0.916503682987639454591301867569, 2.70409395048610954302247767942, 4.01779117330008907272035435701, 4.94187462338222078676214496225, 5.80089372663713311830488894593, 6.29924310809690384186277431417, 7.82211545259042145763277400718, 8.451692828459489681706156630866, 9.615332173891233347377760781196, 10.50131596741989460289231785199

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