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Properties

Label 2-1690-5.4-c1-0-23
Degree $2$
Conductor $1690$
Sign $-0.203 - 0.979i$
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 − 0.666i·3-s − 4-s + (2.18 − 0.454i)5-s + 0.666·6-s + 2.41i·7-s − i·8-s + 2.55·9-s + (0.454 + 2.18i)10-s − 1.46·11-s + 0.666i·12-s − 2.41·14-s + (−0.302 − 1.45i)15-s + 16-s + 6.74i·17-s + 2.55i·18-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.384i·3-s − 0.5·4-s + (0.979 − 0.203i)5-s + 0.272·6-s + 0.912i·7-s − 0.353i·8-s + 0.851·9-s + (0.143 + 0.692i)10-s − 0.440·11-s + 0.192i·12-s − 0.645·14-s + (−0.0782 − 0.376i)15-s + 0.250·16-s + 1.63i·17-s + 0.602i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.203 - 0.979i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.203 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1690\)    =    \(2 \cdot 5 \cdot 13^{2}\)
Sign: $-0.203 - 0.979i$
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: \((2,\ 1690,\ (\ :1/2),\ -0.203 - 0.979i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.804132739\)
\(L(\frac12)\) \(\approx\) \(1.804132739\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.18 + 0.454i)T \)
13 \( 1 \)
good3 \( 1 + 0.666iT - 3T^{2} \)
7 \( 1 - 2.41iT - 7T^{2} \)
11 \( 1 + 1.46T + 11T^{2} \)
17 \( 1 - 6.74iT - 17T^{2} \)
19 \( 1 + 7.41T + 19T^{2} \)
23 \( 1 - 7.48iT - 23T^{2} \)
29 \( 1 - 0.696T + 29T^{2} \)
31 \( 1 - 1.36T + 31T^{2} \)
37 \( 1 - 9.25iT - 37T^{2} \)
41 \( 1 - 4.24T + 41T^{2} \)
43 \( 1 + 2.38iT - 43T^{2} \)
47 \( 1 + 2.77iT - 47T^{2} \)
53 \( 1 + 7.04iT - 53T^{2} \)
59 \( 1 - 0.0690T + 59T^{2} \)
61 \( 1 - 5.91T + 61T^{2} \)
67 \( 1 + 10.9iT - 67T^{2} \)
71 \( 1 + 3.60T + 71T^{2} \)
73 \( 1 - 9.30iT - 73T^{2} \)
79 \( 1 - 0.766T + 79T^{2} \)
83 \( 1 - 16.5iT - 83T^{2} \)
89 \( 1 + 17.7T + 89T^{2} \)
97 \( 1 - 10.5iT - 97T^{2} \)
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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

−9.524441613498531236667015968399, −8.513830656695063862110707982689, −8.188688803230715969819232077392, −7.00361652023112165150232812467, −6.29083328582694103036602962403, −5.73955844214152086638077350192, −4.87209104717651314144045267404, −3.84459206571364013186337442729, −2.33170406942204297187252319582, −1.50255259839526569912528954135, 0.69935852107848483148856577292, 2.08735739526647091101813023790, 2.90305319971206235340383684329, 4.27991707710052931840353626985, 4.61178041161669136476287635825, 5.77359979155802209098191657871, 6.78999614990177823952476642268, 7.43173407034400393008440821682, 8.638746695560873473204479910962, 9.359451717431954999911772217329

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