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Properties

Label 2-1690-13.12-c1-0-2
Degree $2$
Conductor $1690$
Sign $0.277 - 0.960i$
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 − 2.73·3-s − 4-s + i·5-s + 2.73i·6-s − 3i·7-s + i·8-s + 4.46·9-s + 10-s + 3i·11-s + 2.73·12-s − 3·14-s − 2.73i·15-s + 16-s − 2.19·17-s − 4.46i·18-s + ⋯
L(s)  = 1  − 0.707i·2-s − 1.57·3-s − 0.5·4-s + 0.447i·5-s + 1.11i·6-s − 1.13i·7-s + 0.353i·8-s + 1.48·9-s + 0.316·10-s + 0.904i·11-s + 0.788·12-s − 0.801·14-s − 0.705i·15-s + 0.250·16-s − 0.532·17-s − 1.05i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.277 - 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.277 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1690\)    =    \(2 \cdot 5 \cdot 13^{2}\)
Sign: $0.277 - 0.960i$
Analytic conductor: \(13.4947\)
Root analytic conductor: \(3.67351\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1690} (1351, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1690,\ (\ :1/2),\ 0.277 - 0.960i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2446988465\)
\(L(\frac12)\) \(\approx\) \(0.2446988465\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
13 \( 1 \)
good3 \( 1 + 2.73T + 3T^{2} \)
7 \( 1 + 3iT - 7T^{2} \)
11 \( 1 - 3iT - 11T^{2} \)
17 \( 1 + 2.19T + 17T^{2} \)
19 \( 1 + 6.46iT - 19T^{2} \)
23 \( 1 - 2.53T + 23T^{2} \)
29 \( 1 + 9.46T + 29T^{2} \)
31 \( 1 + 1.26iT - 31T^{2} \)
37 \( 1 + 11.1iT - 37T^{2} \)
41 \( 1 - 10.3iT - 41T^{2} \)
43 \( 1 - 2T + 43T^{2} \)
47 \( 1 + 3iT - 47T^{2} \)
53 \( 1 + 6.46T + 53T^{2} \)
59 \( 1 - 10.3iT - 59T^{2} \)
61 \( 1 + 4.19T + 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 - 6iT - 71T^{2} \)
73 \( 1 - 5.66iT - 73T^{2} \)
79 \( 1 - 6.19T + 79T^{2} \)
83 \( 1 + 2.19iT - 83T^{2} \)
89 \( 1 - 17.1iT - 89T^{2} \)
97 \( 1 - 15.1iT - 97T^{2} \)
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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

−9.760231008536313753608164295079, −9.110052204050806339540623268414, −7.54068632461633091877643968473, −7.10286909771376975736056827330, −6.30217697584504245950318432870, −5.25643816237797744678285066332, −4.54591699765470214921249838028, −3.82562099329964515228277943505, −2.36475958407401519932855221768, −1.01398701873155301359917809772, 0.14318246686547593302648382465, 1.64828296298542587505366964278, 3.43891721525392155422697312019, 4.63854290051627843261950841341, 5.39281929393512979262155437349, 5.89318531108996580182323485630, 6.40147069300902780966816527521, 7.47922325137497181043946126093, 8.406948514731256083285651100791, 9.063915524484536934408756400252

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