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Properties

Label 2-1014-13.12-c1-0-18
Degree $2$
Conductor $1014$
Sign $0.969 + 0.246i$
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  + i·2-s + 3-s − 4-s + 0.356i·5-s + i·6-s − 4.04i·7-s − i·8-s + 9-s − 0.356·10-s + 0.911i·11-s − 12-s + 4.04·14-s + 0.356i·15-s + 16-s + 2.09·17-s + i·18-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.577·3-s − 0.5·4-s + 0.159i·5-s + 0.408i·6-s − 1.53i·7-s − 0.353i·8-s + 0.333·9-s − 0.112·10-s + 0.274i·11-s − 0.288·12-s + 1.08·14-s + 0.0921i·15-s + 0.250·16-s + 0.508·17-s + 0.235i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 + 0.246i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.969 + 0.246i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1014\)    =    \(2 \cdot 3 \cdot 13^{2}\)
Sign: $0.969 + 0.246i$
Analytic conductor: \(8.09683\)
Root analytic conductor: \(2.84549\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1014} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1014,\ (\ :1/2),\ 0.969 + 0.246i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.767054454\)
\(L(\frac12)\) \(\approx\) \(1.767054454\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 - T \)
13 \( 1 \)
good5 \( 1 - 0.356iT - 5T^{2} \)
7 \( 1 + 4.04iT - 7T^{2} \)
11 \( 1 - 0.911iT - 11T^{2} \)
17 \( 1 - 2.09T + 17T^{2} \)
19 \( 1 + 4.98iT - 19T^{2} \)
23 \( 1 + 8.49T + 23T^{2} \)
29 \( 1 - 8.51T + 29T^{2} \)
31 \( 1 + 10.7iT - 31T^{2} \)
37 \( 1 - 0.615iT - 37T^{2} \)
41 \( 1 + 7.60iT - 41T^{2} \)
43 \( 1 - 6.27T + 43T^{2} \)
47 \( 1 + 1.78iT - 47T^{2} \)
53 \( 1 - 10.4T + 53T^{2} \)
59 \( 1 - 6.04iT - 59T^{2} \)
61 \( 1 + 3.10T + 61T^{2} \)
67 \( 1 - 13.5iT - 67T^{2} \)
71 \( 1 - 11.4iT - 71T^{2} \)
73 \( 1 + 0.533iT - 73T^{2} \)
79 \( 1 + 11.7T + 79T^{2} \)
83 \( 1 + 6.49iT - 83T^{2} \)
89 \( 1 - 6.49iT - 89T^{2} \)
97 \( 1 + 1.96iT - 97T^{2} \)
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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

−10.04550144665846636610389968468, −8.951411895162805257418433773131, −8.102912983651598177842047325708, −7.33782236418938782177956186961, −6.85643213529482731018323211833, −5.74630132582370087513351445432, −4.41410185975906623269481760022, −3.97449162334367827988991945558, −2.59293991548392651737847203252, −0.818463950584149866922661104069, 1.51694862897657176102182033136, 2.61392743240923314293936447157, 3.40510958822749639458182170649, 4.63236667955651995261927928729, 5.59482421480915342104745813172, 6.43653169001548234022241183325, 7.949059274963018567982878456256, 8.459800095926178582874987585792, 9.132745109813971711842790539990, 9.989948169062189926226391591278

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