Properties

Label 2-690-5.4-c1-0-19
Degree $2$
Conductor $690$
Sign $-0.948 + 0.316i$
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 + (−0.707 − 2.12i)5-s + 6-s − 2i·7-s i·8-s − 9-s + (2.12 − 0.707i)10-s − 4.24·11-s + i·12-s + 0.828i·13-s + 2·14-s + (−2.12 + 0.707i)15-s + 16-s + 6.82i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + (−0.316 − 0.948i)5-s + 0.408·6-s − 0.755i·7-s − 0.353i·8-s − 0.333·9-s + (0.670 − 0.223i)10-s − 1.27·11-s + 0.288i·12-s + 0.229i·13-s + 0.534·14-s + (−0.547 + 0.182i)15-s + 0.250·16-s + 1.65i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.948 + 0.316i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.948 + 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.948 + 0.316i$
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: \((2,\ 690,\ (\ :1/2),\ -0.948 + 0.316i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0480295 - 0.295971i\)
\(L(\frac12)\) \(\approx\) \(0.0480295 - 0.295971i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 + 2.12i)T \)
23 \( 1 - iT \)
good7 \( 1 + 2iT - 7T^{2} \)
11 \( 1 + 4.24T + 11T^{2} \)
13 \( 1 - 0.828iT - 13T^{2} \)
17 \( 1 - 6.82iT - 17T^{2} \)
19 \( 1 + 6.24T + 19T^{2} \)
29 \( 1 + 3.65T + 29T^{2} \)
31 \( 1 - 6T + 31T^{2} \)
37 \( 1 - 0.585iT - 37T^{2} \)
41 \( 1 + 6.82T + 41T^{2} \)
43 \( 1 + 10.2iT - 43T^{2} \)
47 \( 1 + 0.828iT - 47T^{2} \)
53 \( 1 + 10.5iT - 53T^{2} \)
59 \( 1 + 8.48T + 59T^{2} \)
61 \( 1 + 0.585T + 61T^{2} \)
67 \( 1 + 3.41iT - 67T^{2} \)
71 \( 1 + 5.65T + 71T^{2} \)
73 \( 1 + 7.65iT - 73T^{2} \)
79 \( 1 - 3.65T + 79T^{2} \)
83 \( 1 + 1.41iT - 83T^{2} \)
89 \( 1 + 9.17T + 89T^{2} \)
97 \( 1 + 6iT - 97T^{2} \)
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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

−10.13274255174500266859779060170, −8.782120607035016072054041798666, −8.242227452356724654053338419119, −7.59037229561093858775508131553, −6.58169695763093165944208686867, −5.65294831464020631034018359279, −4.65514637526271805955202461886, −3.73045055904861672129954303485, −1.82728893650453217587657874478, −0.14945565433026806110664592980, 2.51405555827995043101920717066, 2.94488202302456260376400984208, 4.33194506087654234908927267274, 5.24179012591949861526600923519, 6.29298581092539364957621248367, 7.54742181514054021058892365220, 8.390349576715508357421841945618, 9.352916556719555667384233896954, 10.19947617206147610106442181327, 10.79302885299820121484130424615

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