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Properties

Label 2-1690-5.4-c1-0-48
Degree $2$
Conductor $1690$
Sign $0.0717 + 0.997i$
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 − 3.06i·3-s − 4-s + (−2.23 + 0.160i)5-s + 3.06·6-s + 3.06i·7-s − i·8-s − 6.41·9-s + (−0.160 − 2.23i)10-s + 5.86·11-s + 3.06i·12-s − 3.06·14-s + (0.492 + 6.84i)15-s + 16-s − 3.26i·17-s − 6.41i·18-s + ⋯
L(s)  = 1  + 0.707i·2-s − 1.77i·3-s − 0.5·4-s + (−0.997 + 0.0717i)5-s + 1.25·6-s + 1.15i·7-s − 0.353i·8-s − 2.13·9-s + (−0.0507 − 0.705i)10-s + 1.76·11-s + 0.885i·12-s − 0.819·14-s + (0.127 + 1.76i)15-s + 0.250·16-s − 0.791i·17-s − 1.51i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0717 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0717 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1690\)    =    \(2 \cdot 5 \cdot 13^{2}\)
Sign: $0.0717 + 0.997i$
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: \((2,\ 1690,\ (\ :1/2),\ 0.0717 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.129577221\)
\(L(\frac12)\) \(\approx\) \(1.129577221\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.23 - 0.160i)T \)
13 \( 1 \)
good3 \( 1 + 3.06iT - 3T^{2} \)
7 \( 1 - 3.06iT - 7T^{2} \)
11 \( 1 - 5.86T + 11T^{2} \)
17 \( 1 + 3.26iT - 17T^{2} \)
19 \( 1 - 0.146T + 19T^{2} \)
23 \( 1 + 0.0169iT - 23T^{2} \)
29 \( 1 - 2.35T + 29T^{2} \)
31 \( 1 + 5.33T + 31T^{2} \)
37 \( 1 + 10.2iT - 37T^{2} \)
41 \( 1 - 2.25T + 41T^{2} \)
43 \( 1 + 6.21iT - 43T^{2} \)
47 \( 1 - 4.28iT - 47T^{2} \)
53 \( 1 + 5.54iT - 53T^{2} \)
59 \( 1 - 11.2T + 59T^{2} \)
61 \( 1 + 4.21T + 61T^{2} \)
67 \( 1 + 9.43iT - 67T^{2} \)
71 \( 1 + 11.2T + 71T^{2} \)
73 \( 1 - 3.17iT - 73T^{2} \)
79 \( 1 + 4.50T + 79T^{2} \)
83 \( 1 - 1.11iT - 83T^{2} \)
89 \( 1 + 7.13T + 89T^{2} \)
97 \( 1 + 10.0iT - 97T^{2} \)
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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

−8.797951304502296731906087444037, −8.300585895265005052896387568581, −7.22267988516346420064372893439, −7.06696444102135610967125628239, −6.12981475259268778281492436991, −5.47277508398918123146477188886, −4.14918996234528133437602294912, −3.02321875976477758539985270399, −1.81843109692839822937030479486, −0.52384560214269815444622559028, 1.13418953932730081021989468177, 3.13743367004815535426466267968, 3.89645302201925280118615636314, 4.14721094262572419140316220684, 4.89832243474602308855365532547, 6.18359118440679673065894435769, 7.24059004843708973303273104167, 8.381239526468770560536589459574, 8.905038817137682223818790864794, 9.725600185675594530948098728052

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