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Properties

Label 2-1690-5.4-c1-0-4
Degree $2$
Conductor $1690$
Sign $-0.763 + 0.645i$
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 + 3.41i·3-s − 4-s + (1.44 + 1.70i)5-s + 3.41·6-s + 0.459i·7-s + i·8-s − 8.66·9-s + (1.70 − 1.44i)10-s − 2.03·11-s − 3.41i·12-s + 0.459·14-s + (−5.83 + 4.92i)15-s + 16-s − 5.39i·17-s + 8.66i·18-s + ⋯
L(s)  = 1  − 0.707i·2-s + 1.97i·3-s − 0.5·4-s + (0.645 + 0.763i)5-s + 1.39·6-s + 0.173i·7-s + 0.353i·8-s − 2.88·9-s + (0.540 − 0.456i)10-s − 0.612·11-s − 0.986i·12-s + 0.122·14-s + (−1.50 + 1.27i)15-s + 0.250·16-s − 1.30i·17-s + 2.04i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.763 + 0.645i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.763 + 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1690\)    =    \(2 \cdot 5 \cdot 13^{2}\)
Sign: $-0.763 + 0.645i$
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: \((2,\ 1690,\ (\ :1/2),\ -0.763 + 0.645i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4355706934\)
\(L(\frac12)\) \(\approx\) \(0.4355706934\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.41iT - 3T^{2} \)
7 \( 1 - 0.459iT - 7T^{2} \)
11 \( 1 + 2.03T + 11T^{2} \)
17 \( 1 + 5.39iT - 17T^{2} \)
19 \( 1 + 3.30T + 19T^{2} \)
23 \( 1 + 4.15iT - 23T^{2} \)
29 \( 1 + 6.05T + 29T^{2} \)
31 \( 1 + 6.52T + 31T^{2} \)
37 \( 1 - 8.07iT - 37T^{2} \)
41 \( 1 - 4.14T + 41T^{2} \)
43 \( 1 - 1.26iT - 43T^{2} \)
47 \( 1 - 1.29iT - 47T^{2} \)
53 \( 1 - 5.78iT - 53T^{2} \)
59 \( 1 + 6.77T + 59T^{2} \)
61 \( 1 + 5.98T + 61T^{2} \)
67 \( 1 + 8.41iT - 67T^{2} \)
71 \( 1 - 1.47T + 71T^{2} \)
73 \( 1 - 8.05iT - 73T^{2} \)
79 \( 1 - 1.68T + 79T^{2} \)
83 \( 1 - 9.99iT - 83T^{2} \)
89 \( 1 - 3.93T + 89T^{2} \)
97 \( 1 - 7.05iT - 97T^{2} \)
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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

−9.873980373176991532967883587453, −9.331720344362874236287842122623, −8.780757142160272614907233687748, −7.66040548131060122771411062742, −6.27859196976184238689314279783, −5.44166007841185616654689242691, −4.81164302446503256134314272084, −3.89689492226917566912105779688, −2.97090946825688350914668373668, −2.40066279633254712203900177416, 0.15483989264099229879947520316, 1.51464898330310025867480988673, 2.24188648682255049732080011580, 3.77367209052551314274156771464, 5.27295856553908838297268511122, 5.83866552614595617531140995832, 6.38507317873163266333164580937, 7.48358954290417183818432466972, 7.73394924940795663844319367474, 8.739552219071676935417668001304

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