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Properties

Label 2-1014-39.8-c1-0-21
Degree $2$
Conductor $1014$
Sign $0.566 + 0.824i$
Analytic cond. $8.09683$
Root an. cond. $2.84549$
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  + (0.707 − 0.707i)2-s + (−1.53 − 0.796i)3-s − 1.00i·4-s + (−0.428 + 0.428i)5-s + (−1.65 + 0.524i)6-s + (0.538 − 0.538i)7-s + (−0.707 − 0.707i)8-s + (1.73 + 2.44i)9-s + 0.606i·10-s + (2.97 + 2.97i)11-s + (−0.796 + 1.53i)12-s − 0.761i·14-s + (1.00 − 0.317i)15-s − 1.00·16-s + 5.24·17-s + (2.95 + 0.507i)18-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (−0.888 − 0.459i)3-s − 0.500i·4-s + (−0.191 + 0.191i)5-s + (−0.673 + 0.214i)6-s + (0.203 − 0.203i)7-s + (−0.250 − 0.250i)8-s + (0.577 + 0.816i)9-s + 0.191i·10-s + (0.895 + 0.895i)11-s + (−0.229 + 0.444i)12-s − 0.203i·14-s + (0.258 − 0.0820i)15-s − 0.250·16-s + 1.27·17-s + (0.696 + 0.119i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.566 + 0.824i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.566 + 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1014\)    =    \(2 \cdot 3 \cdot 13^{2}\)
Sign: $0.566 + 0.824i$
Analytic conductor: \(8.09683\)
Root analytic conductor: \(2.84549\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1014} (437, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1014,\ (\ :1/2),\ 0.566 + 0.824i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.661213144\)
\(L(\frac12)\) \(\approx\) \(1.661213144\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 0.707i)T \)
3 \( 1 + (1.53 + 0.796i)T \)
13 \( 1 \)
good5 \( 1 + (0.428 - 0.428i)T - 5iT^{2} \)
7 \( 1 + (-0.538 + 0.538i)T - 7iT^{2} \)
11 \( 1 + (-2.97 - 2.97i)T + 11iT^{2} \)
17 \( 1 - 5.24T + 17T^{2} \)
19 \( 1 + (-2.41 - 2.41i)T + 19iT^{2} \)
23 \( 1 + 1.86T + 23T^{2} \)
29 \( 1 + 8.70iT - 29T^{2} \)
31 \( 1 + (2.68 + 2.68i)T + 31iT^{2} \)
37 \( 1 + (-4.15 + 4.15i)T - 37iT^{2} \)
41 \( 1 + (-6.27 + 6.27i)T - 41iT^{2} \)
43 \( 1 - 1.95iT - 43T^{2} \)
47 \( 1 + (-5.73 - 5.73i)T + 47iT^{2} \)
53 \( 1 + 9.01iT - 53T^{2} \)
59 \( 1 + (-6.10 - 6.10i)T + 59iT^{2} \)
61 \( 1 + 8.13T + 61T^{2} \)
67 \( 1 + (-0.0740 - 0.0740i)T + 67iT^{2} \)
71 \( 1 + (-7.37 + 7.37i)T - 71iT^{2} \)
73 \( 1 + (-5.57 + 5.57i)T - 73iT^{2} \)
79 \( 1 + 13.5T + 79T^{2} \)
83 \( 1 + (-0.996 + 0.996i)T - 83iT^{2} \)
89 \( 1 + (-4.62 - 4.62i)T + 89iT^{2} \)
97 \( 1 + (-11.1 - 11.1i)T + 97iT^{2} \)
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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

−9.954110992608742502934172083823, −9.389185605122428194085198965133, −7.72800129710300564049947116037, −7.38197965393161075023712692919, −6.17496299408623218645912483341, −5.59371889243694879101020743078, −4.47905082472493799587064293626, −3.71567985962737581321678507712, −2.14674151347200486325235640368, −1.03637196412622043667808900088, 1.03993884516525700375301931835, 3.17107594599887901604432339178, 4.03204178762658620909374654829, 5.02543609114431231447655909654, 5.71054682218732585771024552020, 6.48916140434083865471529008095, 7.36836613181497867826562612124, 8.438312225317635902666141379589, 9.194432795652422128777281585211, 10.13670132256135831009595652819

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