Properties

Label 2-690-15.8-c1-0-24
Degree $2$
Conductor $690$
Sign $-0.856 + 0.515i$
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  + (−0.707 − 0.707i)2-s + (−1.51 − 0.838i)3-s + 1.00i·4-s + (−2.23 + 0.0661i)5-s + (0.479 + 1.66i)6-s + (2.39 − 2.39i)7-s + (0.707 − 0.707i)8-s + (1.59 + 2.54i)9-s + (1.62 + 1.53i)10-s − 3.77i·11-s + (0.838 − 1.51i)12-s + (3.87 + 3.87i)13-s − 3.38·14-s + (3.44 + 1.77i)15-s − 1.00·16-s + (2.58 + 2.58i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (−0.875 − 0.483i)3-s + 0.500i·4-s + (−0.999 + 0.0295i)5-s + (0.195 + 0.679i)6-s + (0.905 − 0.905i)7-s + (0.250 − 0.250i)8-s + (0.531 + 0.846i)9-s + (0.514 + 0.484i)10-s − 1.13i·11-s + (0.241 − 0.437i)12-s + (1.07 + 1.07i)13-s − 0.905·14-s + (0.889 + 0.457i)15-s − 0.250·16-s + (0.626 + 0.626i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.856 + 0.515i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.856 + 0.515i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.856 + 0.515i$
Analytic conductor: \(5.50967\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1/2),\ -0.856 + 0.515i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.170662 - 0.614568i\)
\(L(\frac12)\) \(\approx\) \(0.170662 - 0.614568i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 0.707i)T \)
3 \( 1 + (1.51 + 0.838i)T \)
5 \( 1 + (2.23 - 0.0661i)T \)
23 \( 1 + (0.707 - 0.707i)T \)
good7 \( 1 + (-2.39 + 2.39i)T - 7iT^{2} \)
11 \( 1 + 3.77iT - 11T^{2} \)
13 \( 1 + (-3.87 - 3.87i)T + 13iT^{2} \)
17 \( 1 + (-2.58 - 2.58i)T + 17iT^{2} \)
19 \( 1 + 6.80iT - 19T^{2} \)
29 \( 1 + 6.48T + 29T^{2} \)
31 \( 1 + 7.09T + 31T^{2} \)
37 \( 1 + (-4.68 + 4.68i)T - 37iT^{2} \)
41 \( 1 + 8.03iT - 41T^{2} \)
43 \( 1 + (2.86 + 2.86i)T + 43iT^{2} \)
47 \( 1 + (3.40 + 3.40i)T + 47iT^{2} \)
53 \( 1 + (-1.15 + 1.15i)T - 53iT^{2} \)
59 \( 1 + 0.993T + 59T^{2} \)
61 \( 1 + 11.6T + 61T^{2} \)
67 \( 1 + (-1.09 + 1.09i)T - 67iT^{2} \)
71 \( 1 - 9.26iT - 71T^{2} \)
73 \( 1 + (-2.14 - 2.14i)T + 73iT^{2} \)
79 \( 1 + 4.59iT - 79T^{2} \)
83 \( 1 + (-5.20 + 5.20i)T - 83iT^{2} \)
89 \( 1 - 9.56T + 89T^{2} \)
97 \( 1 + (10.5 - 10.5i)T - 97iT^{2} \)
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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

−10.69063225138074392317484668748, −9.134060682656348703884463020636, −8.314534679283945611574824989212, −7.50797005784102107449711542407, −6.86315645916982121176822091604, −5.60651917866857378996855655252, −4.35296163082227586553226640566, −3.61221975522759745268142617310, −1.66373573396706601116580450479, −0.50742543765063571413499760621, 1.41993024504378631451605115336, 3.51096496112705305026026392946, 4.68837436286328034854385112996, 5.45095150538377293495805331594, 6.27335076673915599515908167430, 7.63454774316645863179178345550, 7.991137933356820672974750746122, 9.104533162383959708268624850596, 9.969114695865977323527699484267, 10.85898023454239816313518242769

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