Properties

Label 2-690-23.22-c2-0-7
Degree $2$
Conductor $690$
Sign $-0.292 - 0.956i$
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 − 1.73·3-s + 2.00·4-s + 2.23i·5-s − 2.44·6-s + 11.8i·7-s + 2.82·8-s + 2.99·9-s + 3.16i·10-s − 20.7i·11-s − 3.46·12-s + 7.75·13-s + 16.7i·14-s − 3.87i·15-s + 4.00·16-s + 22.9i·17-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.500·4-s + 0.447i·5-s − 0.408·6-s + 1.68i·7-s + 0.353·8-s + 0.333·9-s + 0.316i·10-s − 1.88i·11-s − 0.288·12-s + 0.596·13-s + 1.19i·14-s − 0.258i·15-s + 0.250·16-s + 1.34i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.292 - 0.956i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.292 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.292 - 0.956i$
Analytic conductor: \(18.8011\)
Root analytic conductor: \(4.33602\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1),\ -0.292 - 0.956i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.941370391\)
\(L(\frac12)\) \(\approx\) \(1.941370391\)
\(L(2)\) 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 \( 1 + 1.73T \)
5 \( 1 - 2.23iT \)
23 \( 1 + (21.9 - 6.71i)T \)
good7 \( 1 - 11.8iT - 49T^{2} \)
11 \( 1 + 20.7iT - 121T^{2} \)
13 \( 1 - 7.75T + 169T^{2} \)
17 \( 1 - 22.9iT - 289T^{2} \)
19 \( 1 - 21.0iT - 361T^{2} \)
29 \( 1 + 38.2T + 841T^{2} \)
31 \( 1 - 33.3T + 961T^{2} \)
37 \( 1 - 49.4iT - 1.36e3T^{2} \)
41 \( 1 + 0.679T + 1.68e3T^{2} \)
43 \( 1 + 7.99iT - 1.84e3T^{2} \)
47 \( 1 + 85.9T + 2.20e3T^{2} \)
53 \( 1 - 66.6iT - 2.80e3T^{2} \)
59 \( 1 - 110.T + 3.48e3T^{2} \)
61 \( 1 - 16.6iT - 3.72e3T^{2} \)
67 \( 1 - 117. iT - 4.48e3T^{2} \)
71 \( 1 - 28.9T + 5.04e3T^{2} \)
73 \( 1 - 31.0T + 5.32e3T^{2} \)
79 \( 1 + 105. iT - 6.24e3T^{2} \)
83 \( 1 + 43.1iT - 6.88e3T^{2} \)
89 \( 1 - 49.1iT - 7.92e3T^{2} \)
97 \( 1 - 69.6iT - 9.40e3T^{2} \)
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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

−10.78602390850579015655941278463, −9.865151796233736988779273686032, −8.516104645548525104365164911740, −8.125543502159958528408692488063, −6.40706077059781157091869027200, −5.92949582326132715034901582883, −5.50508443916114594292491524402, −3.90423585894711930116908698283, −3.04728290189606244751471167692, −1.68790390577231907848665884351, 0.58108335845835827023485576546, 2.00626439978411064749721961129, 3.77095820800583224608239217254, 4.55170619046056657602099870606, 5.13650638336419857845564832578, 6.62096203929268364955976434514, 7.11681823468772044207551049421, 7.898503115527864259129599019007, 9.538136181465717967529247614847, 10.03688238033808531586818247634

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