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Properties

Label 2-690-5.3-c2-0-21
Degree $2$
Conductor $690$
Sign $0.456 - 0.889i$
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 + i)2-s + (1.22 + 1.22i)3-s − 2i·4-s + (3.24 − 3.80i)5-s − 2.44·6-s + (−6.75 + 6.75i)7-s + (2 + 2i)8-s + 2.99i·9-s + (0.558 + 7.04i)10-s + 21.5·11-s + (2.44 − 2.44i)12-s + (−1.08 − 1.08i)13-s − 13.5i·14-s + (8.63 − 0.683i)15-s − 4·16-s + (4.33 − 4.33i)17-s + ⋯
L(s)  = 1  + (−0.5 + 0.5i)2-s + (0.408 + 0.408i)3-s − 0.5i·4-s + (0.649 − 0.760i)5-s − 0.408·6-s + (−0.965 + 0.965i)7-s + (0.250 + 0.250i)8-s + 0.333i·9-s + (0.0558 + 0.704i)10-s + 1.96·11-s + (0.204 − 0.204i)12-s + (−0.0833 − 0.0833i)13-s − 0.965i·14-s + (0.575 − 0.0455i)15-s − 0.250·16-s + (0.255 − 0.255i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.456 - 0.889i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.456 - 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.456 - 0.889i$
Analytic conductor: \(18.8011\)
Root analytic conductor: \(4.33602\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (553, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1),\ 0.456 - 0.889i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.824801803\)
\(L(\frac12)\) \(\approx\) \(1.824801803\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 - i)T \)
3 \( 1 + (-1.22 - 1.22i)T \)
5 \( 1 + (-3.24 + 3.80i)T \)
23 \( 1 + (3.39 + 3.39i)T \)
good7 \( 1 + (6.75 - 6.75i)T - 49iT^{2} \)
11 \( 1 - 21.5T + 121T^{2} \)
13 \( 1 + (1.08 + 1.08i)T + 169iT^{2} \)
17 \( 1 + (-4.33 + 4.33i)T - 289iT^{2} \)
19 \( 1 + 1.61iT - 361T^{2} \)
29 \( 1 - 10.6iT - 841T^{2} \)
31 \( 1 - 21.7T + 961T^{2} \)
37 \( 1 + (18.5 - 18.5i)T - 1.36e3iT^{2} \)
41 \( 1 - 55.4T + 1.68e3T^{2} \)
43 \( 1 + (-45.1 - 45.1i)T + 1.84e3iT^{2} \)
47 \( 1 + (40.0 - 40.0i)T - 2.20e3iT^{2} \)
53 \( 1 + (22.8 + 22.8i)T + 2.80e3iT^{2} \)
59 \( 1 - 45.0iT - 3.48e3T^{2} \)
61 \( 1 - 106.T + 3.72e3T^{2} \)
67 \( 1 + (-62.2 + 62.2i)T - 4.48e3iT^{2} \)
71 \( 1 - 95.7T + 5.04e3T^{2} \)
73 \( 1 + (18.3 + 18.3i)T + 5.32e3iT^{2} \)
79 \( 1 - 74.1iT - 6.24e3T^{2} \)
83 \( 1 + (-29.2 - 29.2i)T + 6.88e3iT^{2} \)
89 \( 1 - 148. iT - 7.92e3T^{2} \)
97 \( 1 + (47.9 - 47.9i)T - 9.40e3iT^{2} \)
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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

−9.744886307789779697778157101555, −9.523011358106763131545788981878, −8.916678800557398350843599545351, −8.121631337569955766668668045247, −6.66228388216221250864427756953, −6.13919971457831350385874874726, −5.11777407143196241202985416653, −3.95867563091618329492405713762, −2.57318965090937561454850606848, −1.15739163455115091120975908912, 0.908515463464929145067956605758, 2.12204371471815755908393958734, 3.42009132969405996650354103715, 3.99983257253082902491681263060, 6.04010661873619556912036713926, 6.78278966562956257992567557905, 7.30915440815686520635141151041, 8.587109747034695233847275781197, 9.548438927190368686591599886140, 9.869524129741454385269254047901

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