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Properties

Label 2-690-15.2-c1-0-40
Degree $2$
Conductor $690$
Sign $-0.484 + 0.874i$
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 + (1.72 + 0.144i)3-s − 1.00i·4-s + (−1.25 − 1.85i)5-s + (1.32 − 1.11i)6-s + (−2.31 − 2.31i)7-s + (−0.707 − 0.707i)8-s + (2.95 + 0.499i)9-s + (−2.19 − 0.423i)10-s + 0.0826i·11-s + (0.144 − 1.72i)12-s + (−2.92 + 2.92i)13-s − 3.27·14-s + (−1.89 − 3.37i)15-s − 1.00·16-s + (5.30 − 5.30i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (0.996 + 0.0834i)3-s − 0.500i·4-s + (−0.560 − 0.828i)5-s + (0.539 − 0.456i)6-s + (−0.875 − 0.875i)7-s + (−0.250 − 0.250i)8-s + (0.986 + 0.166i)9-s + (−0.694 − 0.134i)10-s + 0.0249i·11-s + (0.0417 − 0.498i)12-s + (−0.810 + 0.810i)13-s − 0.875·14-s + (−0.489 − 0.872i)15-s − 0.250·16-s + (1.28 − 1.28i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.484 + 0.874i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.484 + 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.484 + 0.874i$
Analytic conductor: \(5.50967\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1/2),\ -0.484 + 0.874i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.08035 - 1.83398i\)
\(L(\frac12)\) \(\approx\) \(1.08035 - 1.83398i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.72 - 0.144i)T \)
5 \( 1 + (1.25 + 1.85i)T \)
23 \( 1 + (-0.707 - 0.707i)T \)
good7 \( 1 + (2.31 + 2.31i)T + 7iT^{2} \)
11 \( 1 - 0.0826iT - 11T^{2} \)
13 \( 1 + (2.92 - 2.92i)T - 13iT^{2} \)
17 \( 1 + (-5.30 + 5.30i)T - 17iT^{2} \)
19 \( 1 + 6.47iT - 19T^{2} \)
29 \( 1 + 1.98T + 29T^{2} \)
31 \( 1 + 4.16T + 31T^{2} \)
37 \( 1 + (-4.80 - 4.80i)T + 37iT^{2} \)
41 \( 1 - 0.882iT - 41T^{2} \)
43 \( 1 + (1.69 - 1.69i)T - 43iT^{2} \)
47 \( 1 + (-7.63 + 7.63i)T - 47iT^{2} \)
53 \( 1 + (-7.26 - 7.26i)T + 53iT^{2} \)
59 \( 1 - 10.0T + 59T^{2} \)
61 \( 1 + 0.876T + 61T^{2} \)
67 \( 1 + (-0.768 - 0.768i)T + 67iT^{2} \)
71 \( 1 + 1.70iT - 71T^{2} \)
73 \( 1 + (0.265 - 0.265i)T - 73iT^{2} \)
79 \( 1 - 16.8iT - 79T^{2} \)
83 \( 1 + (-8.84 - 8.84i)T + 83iT^{2} \)
89 \( 1 + 5.65T + 89T^{2} \)
97 \( 1 + (7.30 + 7.30i)T + 97iT^{2} \)
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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

−9.822365596196653536137682377871, −9.581774030373452308453812440664, −8.662374205540336628688279772436, −7.33209809021880461289474439767, −7.05347844238741319768807742981, −5.24424839064246375508004602056, −4.39588942830826831958414074672, −3.57582365221363452471911909961, −2.57798466424106806430364952480, −0.871760419167569508661500391818, 2.33955184210843367442289551667, 3.33626467326106327922187276548, 3.88324502653892122288659112634, 5.56419923143437374879018039901, 6.29313082238421555580258799447, 7.49044528352330239097227061309, 7.86651992108631761482745340417, 8.844422500834402978343393515320, 9.909651213978072254307800346545, 10.48011232121912700242821131411

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