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Properties

Label 2-1690-5.4-c1-0-2
Degree $2$
Conductor $1690$
Sign $-0.762 + 0.647i$
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 + 2.89i·3-s − 4-s + (−1.44 − 1.70i)5-s + 2.89·6-s + 4.38i·7-s + i·8-s − 5.38·9-s + (−1.70 + 1.44i)10-s − 2·11-s − 2.89i·12-s + 4.38·14-s + (4.93 − 4.19i)15-s + 16-s + 5.86i·17-s + 5.38i·18-s + ⋯
L(s)  = 1  − 0.707i·2-s + 1.67i·3-s − 0.5·4-s + (−0.647 − 0.762i)5-s + 1.18·6-s + 1.65i·7-s + 0.353i·8-s − 1.79·9-s + (−0.538 + 0.457i)10-s − 0.603·11-s − 0.835i·12-s + 1.17·14-s + (1.27 − 1.08i)15-s + 0.250·16-s + 1.42i·17-s + 1.26i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.762 + 0.647i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.762 + 0.647i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1690\)    =    \(2 \cdot 5 \cdot 13^{2}\)
Sign: $-0.762 + 0.647i$
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: \((2,\ 1690,\ (\ :1/2),\ -0.762 + 0.647i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2113786596\)
\(L(\frac12)\) \(\approx\) \(0.2113786596\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.44 + 1.70i)T \)
13 \( 1 \)
good3 \( 1 - 2.89iT - 3T^{2} \)
7 \( 1 - 4.38iT - 7T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
17 \( 1 - 5.86iT - 17T^{2} \)
19 \( 1 + 0.973T + 19T^{2} \)
23 \( 1 + 7.79iT - 23T^{2} \)
29 \( 1 + 0.973T + 29T^{2} \)
31 \( 1 - 1.79T + 31T^{2} \)
37 \( 1 - 0.591iT - 37T^{2} \)
41 \( 1 + 4.81T + 41T^{2} \)
43 \( 1 + 4.68iT - 43T^{2} \)
47 \( 1 + 0.381iT - 47T^{2} \)
53 \( 1 + 7.79iT - 53T^{2} \)
59 \( 1 + 0.973T + 59T^{2} \)
61 \( 1 + 0.817T + 61T^{2} \)
67 \( 1 + 1.79iT - 67T^{2} \)
71 \( 1 - 3.92T + 71T^{2} \)
73 \( 1 + 6iT - 73T^{2} \)
79 \( 1 + 10.9T + 79T^{2} \)
83 \( 1 + 6.97iT - 83T^{2} \)
89 \( 1 - 0.973T + 89T^{2} \)
97 \( 1 - 18.6iT - 97T^{2} \)
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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

−9.914074236566597210843135056856, −8.946114415799620957042032222124, −8.653518092513455477556272873882, −8.088096963577961629098063731479, −6.20453511187055074093560416033, −5.34301471443174326921875196656, −4.80143269876903844644337905766, −4.00375827052342296479312051294, −3.14602098762121065812877992956, −2.11377932089941415207015971715, 0.087069049613404305233188994865, 1.19682268467266008988831935260, 2.73834397609305040470025598667, 3.69953384113011806359785009108, 4.84089813890087779243335022612, 5.98507417745761487824598796042, 6.85713661852566873615385300314, 7.33422428863886622626706293133, 7.60227027536356145412788662049, 8.324945006823865992465988227717

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