Properties

Label 2-690-15.14-c2-0-8
Degree $2$
Conductor $690$
Sign $-0.198 - 0.980i$
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 + (−1.42 + 2.63i)3-s + 2.00·4-s + (−3.84 + 3.20i)5-s + (2.01 − 3.73i)6-s − 12.7i·7-s − 2.82·8-s + (−4.93 − 7.52i)9-s + (5.43 − 4.52i)10-s + 2.23i·11-s + (−2.85 + 5.27i)12-s + 0.635i·13-s + 17.9i·14-s + (−2.97 − 14.7i)15-s + 4.00·16-s − 9.66·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.475 + 0.879i)3-s + 0.500·4-s + (−0.768 + 0.640i)5-s + (0.336 − 0.622i)6-s − 1.81i·7-s − 0.353·8-s + (−0.547 − 0.836i)9-s + (0.543 − 0.452i)10-s + 0.203i·11-s + (−0.237 + 0.439i)12-s + 0.0489i·13-s + 1.28i·14-s + (−0.198 − 0.980i)15-s + 0.250·16-s − 0.568·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.198 - 0.980i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.198 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.198 - 0.980i$
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: \((2,\ 690,\ (\ :1),\ -0.198 - 0.980i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.5687331141\)
\(L(\frac12)\) \(\approx\) \(0.5687331141\)
\(L(2)\) 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 \( 1 + (1.42 - 2.63i)T \)
5 \( 1 + (3.84 - 3.20i)T \)
23 \( 1 - 4.79T \)
good7 \( 1 + 12.7iT - 49T^{2} \)
11 \( 1 - 2.23iT - 121T^{2} \)
13 \( 1 - 0.635iT - 169T^{2} \)
17 \( 1 + 9.66T + 289T^{2} \)
19 \( 1 - 3.49T + 361T^{2} \)
29 \( 1 - 36.8iT - 841T^{2} \)
31 \( 1 - 38.8T + 961T^{2} \)
37 \( 1 - 2.05iT - 1.36e3T^{2} \)
41 \( 1 + 20.9iT - 1.68e3T^{2} \)
43 \( 1 - 44.3iT - 1.84e3T^{2} \)
47 \( 1 + 12.6T + 2.20e3T^{2} \)
53 \( 1 + 32.1T + 2.80e3T^{2} \)
59 \( 1 - 34.6iT - 3.48e3T^{2} \)
61 \( 1 - 15.4T + 3.72e3T^{2} \)
67 \( 1 - 103. iT - 4.48e3T^{2} \)
71 \( 1 + 103. iT - 5.04e3T^{2} \)
73 \( 1 - 66.4iT - 5.32e3T^{2} \)
79 \( 1 + 62.2T + 6.24e3T^{2} \)
83 \( 1 + 21.9T + 6.88e3T^{2} \)
89 \( 1 - 12.4iT - 7.92e3T^{2} \)
97 \( 1 - 110. iT - 9.40e3T^{2} \)
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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

−10.41322814174145931712956088282, −10.00663131047476889249064314520, −8.875293135498050369980021121569, −7.84427887149401700068348823969, −7.03283196619079570683799558076, −6.41258785994171156760122535751, −4.79089490967689813169269044622, −3.99514312594382673797408289027, −3.07543924076071883203498336737, −0.898484413610325321496079907594, 0.37001809817623300701808697293, 1.85265777927169923824418464537, 2.91446972134936381492544189864, 4.76184720326959482210162612092, 5.72629292143172640700020280395, 6.47265425812263427746585796561, 7.61983249280022511278053831601, 8.402232115351204033038289627977, 8.819139825274532639602502859840, 9.871948990927874700023538972021

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