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Properties

Label 2-690-345.344-c1-0-47
Degree $2$
Conductor $690$
Sign $-0.786 + 0.617i$
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  + 2-s + (0.250 − 1.71i)3-s + 4-s + (−0.545 − 2.16i)5-s + (0.250 − 1.71i)6-s − 1.86·7-s + 8-s + (−2.87 − 0.858i)9-s + (−0.545 − 2.16i)10-s − 2.81·11-s + (0.250 − 1.71i)12-s − 1.42i·13-s − 1.86·14-s + (−3.85 + 0.390i)15-s + 16-s − 3.10i·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.144 − 0.989i)3-s + 0.5·4-s + (−0.243 − 0.969i)5-s + (0.102 − 0.699i)6-s − 0.703·7-s + 0.353·8-s + (−0.958 − 0.286i)9-s + (−0.172 − 0.685i)10-s − 0.849·11-s + (0.0723 − 0.494i)12-s − 0.395i·13-s − 0.497·14-s + (−0.994 + 0.100i)15-s + 0.250·16-s − 0.753i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.786 + 0.617i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.786 + 0.617i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.786 + 0.617i$
Analytic conductor: \(5.50967\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (689, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1/2),\ -0.786 + 0.617i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.559463 - 1.61827i\)
\(L(\frac12)\) \(\approx\) \(0.559463 - 1.61827i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + (-0.250 + 1.71i)T \)
5 \( 1 + (0.545 + 2.16i)T \)
23 \( 1 + (-4.05 + 2.56i)T \)
good7 \( 1 + 1.86T + 7T^{2} \)
11 \( 1 + 2.81T + 11T^{2} \)
13 \( 1 + 1.42iT - 13T^{2} \)
17 \( 1 + 3.10iT - 17T^{2} \)
19 \( 1 - 3.42iT - 19T^{2} \)
29 \( 1 + 6.01iT - 29T^{2} \)
31 \( 1 - 5.09T + 31T^{2} \)
37 \( 1 - 1.27T + 37T^{2} \)
41 \( 1 + 1.76iT - 41T^{2} \)
43 \( 1 - 9.25T + 43T^{2} \)
47 \( 1 + 2.70T + 47T^{2} \)
53 \( 1 + 5.16iT - 53T^{2} \)
59 \( 1 + 5.36iT - 59T^{2} \)
61 \( 1 - 6.07iT - 61T^{2} \)
67 \( 1 - 8.70T + 67T^{2} \)
71 \( 1 + 4.56iT - 71T^{2} \)
73 \( 1 + 1.66iT - 73T^{2} \)
79 \( 1 + 10.4iT - 79T^{2} \)
83 \( 1 + 6.08iT - 83T^{2} \)
89 \( 1 - 8.61T + 89T^{2} \)
97 \( 1 + 7.72T + 97T^{2} \)
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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

−10.15753983361694610837490887201, −9.150843376156770112707420292218, −8.126515101619474112217606820819, −7.55749533419469348730578344755, −6.45564188464634915852036610553, −5.64153109069328752562391900961, −4.72107025117063313194069616258, −3.36751446100260447795193853732, −2.33992698199390406671369261543, −0.67893697360482382035051441461, 2.60591755534897136370866827894, 3.26744609724162021771099519461, 4.24855668310471935481261595324, 5.28049969924191141632365489313, 6.26390065675066652064872277163, 7.11939418633222566189957030155, 8.171791131551907006871582540166, 9.299334357831940289074258069051, 10.13659828684100485965720436771, 10.87988506159746641801873695916

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