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Properties

Label 2-1014-13.12-c3-0-76
Degree $2$
Conductor $1014$
Sign $-0.722 + 0.691i$
Analytic cond. $59.8279$
Root an. cond. $7.73485$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2i·2-s + 3·3-s − 4·4-s − 3.19i·5-s + 6i·6-s − 34.6i·7-s − 8i·8-s + 9·9-s + 6.39·10-s − 63.0i·11-s − 12·12-s + 69.3·14-s − 9.59i·15-s + 16·16-s − 79.6·17-s + 18i·18-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.577·3-s − 0.5·4-s − 0.285i·5-s + 0.408i·6-s − 1.87i·7-s − 0.353i·8-s + 0.333·9-s + 0.202·10-s − 1.72i·11-s − 0.288·12-s + 1.32·14-s − 0.165i·15-s + 0.250·16-s − 1.13·17-s + 0.235i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.722 + 0.691i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.722 + 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1014\)    =    \(2 \cdot 3 \cdot 13^{2}\)
Sign: $-0.722 + 0.691i$
Analytic conductor: \(59.8279\)
Root analytic conductor: \(7.73485\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1014} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1014,\ (\ :3/2),\ -0.722 + 0.691i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.269620033\)
\(L(\frac12)\) \(\approx\) \(1.269620033\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2iT \)
3 \( 1 - 3T \)
13 \( 1 \)
good5 \( 1 + 3.19iT - 125T^{2} \)
7 \( 1 + 34.6iT - 343T^{2} \)
11 \( 1 + 63.0iT - 1.33e3T^{2} \)
17 \( 1 + 79.6T + 4.91e3T^{2} \)
19 \( 1 - 35.0iT - 6.85e3T^{2} \)
23 \( 1 - 67.4T + 1.21e4T^{2} \)
29 \( 1 + 247.T + 2.43e4T^{2} \)
31 \( 1 + 141. iT - 2.97e4T^{2} \)
37 \( 1 - 263. iT - 5.06e4T^{2} \)
41 \( 1 + 22.9iT - 6.89e4T^{2} \)
43 \( 1 + 377.T + 7.95e4T^{2} \)
47 \( 1 - 336. iT - 1.03e5T^{2} \)
53 \( 1 - 543.T + 1.48e5T^{2} \)
59 \( 1 - 114. iT - 2.05e5T^{2} \)
61 \( 1 + 789.T + 2.26e5T^{2} \)
67 \( 1 - 707. iT - 3.00e5T^{2} \)
71 \( 1 + 428. iT - 3.57e5T^{2} \)
73 \( 1 - 403. iT - 3.89e5T^{2} \)
79 \( 1 - 311.T + 4.93e5T^{2} \)
83 \( 1 + 94.5iT - 5.71e5T^{2} \)
89 \( 1 + 711. iT - 7.04e5T^{2} \)
97 \( 1 + 44.1iT - 9.12e5T^{2} \)
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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

−9.000897569649428774683025471124, −8.387506954938169902988375815207, −7.57431839698127185593506237020, −6.87398635016864939989178032924, −5.99957597895296578476416723286, −4.77026484171157180659298428079, −3.95229355318741100015240985584, −3.16229022699312116292898061689, −1.28059393415680991288091207513, −0.29039209086528158938578960914, 1.91408713910766010580732749279, 2.30884608212862595087776990522, 3.34228362858785389469859930358, 4.63858085906899478183040385901, 5.30360799889398584258861768556, 6.60549504705343440885725738889, 7.42353726623986605627936858021, 8.681221887215644179749230201586, 9.056258005074757002914829414788, 9.709426895669889298333095304614

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