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Properties

Label 2-690-15.14-c2-0-25
Degree $2$
Conductor $690$
Sign $0.764 - 0.644i$
Analytic cond. $18.8011$
Root an. cond. $4.33602$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41·2-s + (−2.79 + 1.07i)3-s + 2.00·4-s + (4.72 − 1.63i)5-s + (3.95 − 1.52i)6-s − 7.69i·7-s − 2.82·8-s + (6.67 − 6.03i)9-s + (−6.68 + 2.31i)10-s + 16.6i·11-s + (−5.59 + 2.15i)12-s + 13.7i·13-s + 10.8i·14-s + (−11.4 + 9.67i)15-s + 4.00·16-s − 26.8·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.933 + 0.359i)3-s + 0.500·4-s + (0.945 − 0.326i)5-s + (0.659 − 0.254i)6-s − 1.09i·7-s − 0.353·8-s + (0.741 − 0.671i)9-s + (−0.668 + 0.231i)10-s + 1.51i·11-s + (−0.466 + 0.179i)12-s + 1.06i·13-s + 0.777i·14-s + (−0.764 + 0.644i)15-s + 0.250·16-s − 1.58·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.764 - 0.644i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.764 - 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.764 - 0.644i$
Analytic conductor: \(18.8011\)
Root analytic conductor: \(4.33602\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (599, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1),\ 0.764 - 0.644i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.042495048\)
\(L(\frac12)\) \(\approx\) \(1.042495048\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41T \)
3 \( 1 + (2.79 - 1.07i)T \)
5 \( 1 + (-4.72 + 1.63i)T \)
23 \( 1 + 4.79T \)
good7 \( 1 + 7.69iT - 49T^{2} \)
11 \( 1 - 16.6iT - 121T^{2} \)
13 \( 1 - 13.7iT - 169T^{2} \)
17 \( 1 + 26.8T + 289T^{2} \)
19 \( 1 - 31.7T + 361T^{2} \)
29 \( 1 + 26.0iT - 841T^{2} \)
31 \( 1 + 7.68T + 961T^{2} \)
37 \( 1 + 39.6iT - 1.36e3T^{2} \)
41 \( 1 - 64.5iT - 1.68e3T^{2} \)
43 \( 1 - 58.0iT - 1.84e3T^{2} \)
47 \( 1 - 72.2T + 2.20e3T^{2} \)
53 \( 1 - 20.7T + 2.80e3T^{2} \)
59 \( 1 + 30.3iT - 3.48e3T^{2} \)
61 \( 1 + 7.15T + 3.72e3T^{2} \)
67 \( 1 - 88.3iT - 4.48e3T^{2} \)
71 \( 1 - 71.2iT - 5.04e3T^{2} \)
73 \( 1 + 28.2iT - 5.32e3T^{2} \)
79 \( 1 + 23.1T + 6.24e3T^{2} \)
83 \( 1 - 139.T + 6.88e3T^{2} \)
89 \( 1 - 44.9iT - 7.92e3T^{2} \)
97 \( 1 + 59.4iT - 9.40e3T^{2} \)
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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

−10.13196713844615858215893456885, −9.666905212833369857263349447812, −9.079055014212526393415564652863, −7.46365800260563005720332125320, −6.91077529015573083318933356650, −6.06868575168404347491854893491, −4.81538755798150035924756095985, −4.16884085715304201861160179609, −2.13115027848371556791177085065, −1.00802721399471873450497867775, 0.66532983004726491821349772166, 2.06993833658016561503632993905, 3.14728318377269993596393534453, 5.33590013233684794555503914581, 5.67778825168618306063373473552, 6.52672568610081077754172339929, 7.46854852147487286749238398403, 8.664618001315574070582405280080, 9.197187118965337987582389972715, 10.41093298069372484793438301942

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