Properties

Label 2-690-5.4-c1-0-2
Degree $2$
Conductor $690$
Sign $0.662 - 0.749i$
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  i·2-s i·3-s − 4-s + (−1.67 − 1.48i)5-s − 6-s + 2.96i·7-s + i·8-s − 9-s + (−1.48 + 1.67i)10-s − 3.35·11-s + i·12-s + 4.96i·13-s + 2.96·14-s + (−1.48 + 1.67i)15-s + 16-s + 1.35i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.577i·3-s − 0.5·4-s + (−0.749 − 0.662i)5-s − 0.408·6-s + 1.11i·7-s + 0.353i·8-s − 0.333·9-s + (−0.468 + 0.529i)10-s − 1.01·11-s + 0.288i·12-s + 1.37i·13-s + 0.791·14-s + (−0.382 + 0.432i)15-s + 0.250·16-s + 0.327i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.662 - 0.749i$
Analytic conductor: \(5.50967\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1/2),\ 0.662 - 0.749i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.537578 + 0.242251i\)
\(L(\frac12)\) \(\approx\) \(0.537578 + 0.242251i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 + iT \)
5 \( 1 + (1.67 + 1.48i)T \)
23 \( 1 - iT \)
good7 \( 1 - 2.96iT - 7T^{2} \)
11 \( 1 + 3.35T + 11T^{2} \)
13 \( 1 - 4.96iT - 13T^{2} \)
17 \( 1 - 1.35iT - 17T^{2} \)
19 \( 1 - 4.96T + 19T^{2} \)
29 \( 1 + 7.73T + 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + 7.61iT - 37T^{2} \)
41 \( 1 - 4.70T + 41T^{2} \)
43 \( 1 - 10.3iT - 43T^{2} \)
47 \( 1 - 3.22iT - 47T^{2} \)
53 \( 1 - 6.96iT - 53T^{2} \)
59 \( 1 + 1.22T + 59T^{2} \)
61 \( 1 + 11.0T + 61T^{2} \)
67 \( 1 + 7.61iT - 67T^{2} \)
71 \( 1 + 2.18T + 71T^{2} \)
73 \( 1 - 9.92iT - 73T^{2} \)
79 \( 1 - 4.12T + 79T^{2} \)
83 \( 1 - 6.38iT - 83T^{2} \)
89 \( 1 + 9.92T + 89T^{2} \)
97 \( 1 - 12.8iT - 97T^{2} \)
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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

−10.97847797193717029031664038703, −9.329395430959824711992303907421, −9.159980262017052137001948087874, −7.999080136219275650084962508307, −7.39010595834352775976338951518, −5.88160987938398970056839817903, −5.11996696825022873012102947786, −3.97279167040471620375744686712, −2.71571553421871008147453122111, −1.57517351925560974559097061736, 0.31488607278915811316398700563, 3.04720196881521653527097991728, 3.80224340879444563616649041099, 4.94530519209386311746993164876, 5.76392117959341146464651159109, 7.17981994587165152437397551930, 7.56120688088907304674627686932, 8.338450402531031311380631667170, 9.611605931127727491480621860268, 10.44482545388560738436856991168

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