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Properties

Label 2-1690-5.4-c1-0-57
Degree $2$
Conductor $1690$
Sign $-0.977 + 0.210i$
Analytic cond. $13.4947$
Root an. cond. $3.67351$
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 − 2.52i·3-s − 4-s + (0.469 + 2.18i)5-s − 2.52·6-s + 2.37i·7-s + i·8-s − 3.37·9-s + (2.18 − 0.469i)10-s − 1.58·11-s + 2.52i·12-s + 2.37·14-s + (5.51 − 1.18i)15-s + 16-s − 5.98i·17-s + 3.37i·18-s + ⋯
L(s)  = 1  − 0.707i·2-s − 1.45i·3-s − 0.5·4-s + (0.210 + 0.977i)5-s − 1.03·6-s + 0.896i·7-s + 0.353i·8-s − 1.12·9-s + (0.691 − 0.148i)10-s − 0.477·11-s + 0.728i·12-s + 0.634·14-s + (1.42 − 0.306i)15-s + 0.250·16-s − 1.45i·17-s + 0.794i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.977 + 0.210i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.977 + 0.210i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1690\)    =    \(2 \cdot 5 \cdot 13^{2}\)
Sign: $-0.977 + 0.210i$
Analytic conductor: \(13.4947\)
Root analytic conductor: \(3.67351\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1690} (339, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1690,\ (\ :1/2),\ -0.977 + 0.210i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.231999237\)
\(L(\frac12)\) \(\approx\) \(1.231999237\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
5 \( 1 + (-0.469 - 2.18i)T \)
13 \( 1 \)
good3 \( 1 + 2.52iT - 3T^{2} \)
7 \( 1 - 2.37iT - 7T^{2} \)
11 \( 1 + 1.58T + 11T^{2} \)
17 \( 1 + 5.98iT - 17T^{2} \)
19 \( 1 - 3.46T + 19T^{2} \)
23 \( 1 + 6.63iT - 23T^{2} \)
29 \( 1 - 2.74T + 29T^{2} \)
31 \( 1 + 3.46T + 31T^{2} \)
37 \( 1 + 9.11iT - 37T^{2} \)
41 \( 1 + 10.0T + 41T^{2} \)
43 \( 1 - 0.644iT - 43T^{2} \)
47 \( 1 + 10.3iT - 47T^{2} \)
53 \( 1 + 5.04iT - 53T^{2} \)
59 \( 1 - 6.63T + 59T^{2} \)
61 \( 1 + 6.74T + 61T^{2} \)
67 \( 1 + 4iT - 67T^{2} \)
71 \( 1 - 12.6T + 71T^{2} \)
73 \( 1 - 10iT - 73T^{2} \)
79 \( 1 - 4.74T + 79T^{2} \)
83 \( 1 + 8.74iT - 83T^{2} \)
89 \( 1 - 1.87T + 89T^{2} \)
97 \( 1 + 6.74iT - 97T^{2} \)
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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

−8.943011270692973378509186840850, −8.158255623721101552795189608219, −7.26329801420401946975672988467, −6.76539309025330617208831595176, −5.77969419213408344826527467539, −5.03115084811912059642402936037, −3.40358766286453853302425057678, −2.50588658403738944969464110959, −2.06688548684577528352199527012, −0.48995254482038308987657870019, 1.34138023488007654016971326811, 3.39352502360469439332703094663, 4.04201962428114781255162470072, 4.87536431101210529316421388515, 5.38780713459781740694367449037, 6.31579385805556980159510685655, 7.53257742662798366878499970874, 8.199777249271678834150133559714, 8.989645067172018217472359048980, 9.707336974346921194994189098109

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