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Properties

Label 2-1014-39.5-c1-0-41
Degree $2$
Conductor $1014$
Sign $0.939 + 0.343i$
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 + (1.29 − 1.15i)3-s + 1.00i·4-s + (−1.82 − 1.82i)5-s + (1.72 + 0.0980i)6-s + (2.63 + 2.63i)7-s + (−0.707 + 0.707i)8-s + (0.339 − 2.98i)9-s − 2.58i·10-s + (2.30 − 2.30i)11-s + (1.15 + 1.29i)12-s + 3.72i·14-s + (−4.46 − 0.253i)15-s − 1.00·16-s − 1.34·17-s + (2.34 − 1.86i)18-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (0.746 − 0.665i)3-s + 0.500i·4-s + (−0.817 − 0.817i)5-s + (0.705 + 0.0400i)6-s + (0.994 + 0.994i)7-s + (−0.250 + 0.250i)8-s + (0.113 − 0.993i)9-s − 0.817i·10-s + (0.695 − 0.695i)11-s + (0.332 + 0.373i)12-s + 0.994i·14-s + (−1.15 − 0.0654i)15-s − 0.250·16-s − 0.326·17-s + (0.553 − 0.440i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.343i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 + 0.343i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1014\)    =    \(2 \cdot 3 \cdot 13^{2}\)
Sign: $0.939 + 0.343i$
Analytic conductor: \(8.09683\)
Root analytic conductor: \(2.84549\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1014} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1014,\ (\ :1/2),\ 0.939 + 0.343i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.673477602\)
\(L(\frac12)\) \(\approx\) \(2.673477602\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.29 + 1.15i)T \)
13 \( 1 \)
good5 \( 1 + (1.82 + 1.82i)T + 5iT^{2} \)
7 \( 1 + (-2.63 - 2.63i)T + 7iT^{2} \)
11 \( 1 + (-2.30 + 2.30i)T - 11iT^{2} \)
17 \( 1 + 1.34T + 17T^{2} \)
19 \( 1 + (-3.58 + 3.58i)T - 19iT^{2} \)
23 \( 1 - 3.65T + 23T^{2} \)
29 \( 1 - 3.65iT - 29T^{2} \)
31 \( 1 + (0.321 - 0.321i)T - 31iT^{2} \)
37 \( 1 + (1.95 + 1.95i)T + 37iT^{2} \)
41 \( 1 + (-7.89 - 7.89i)T + 41iT^{2} \)
43 \( 1 + 9.10iT - 43T^{2} \)
47 \( 1 + (4.72 - 4.72i)T - 47iT^{2} \)
53 \( 1 - 0.216iT - 53T^{2} \)
59 \( 1 + (-3.65 + 3.65i)T - 59iT^{2} \)
61 \( 1 - 6.52T + 61T^{2} \)
67 \( 1 + (2.26 - 2.26i)T - 67iT^{2} \)
71 \( 1 + (0.108 + 0.108i)T + 71iT^{2} \)
73 \( 1 + (-3.58 - 3.58i)T + 73iT^{2} \)
79 \( 1 - 4.09T + 79T^{2} \)
83 \( 1 + (10.5 + 10.5i)T + 83iT^{2} \)
89 \( 1 + (8.85 - 8.85i)T - 89iT^{2} \)
97 \( 1 + (0.168 - 0.168i)T - 97iT^{2} \)
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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

−9.328575450360517320774284859956, −8.760842347496023731880226239715, −8.307053718900553770376593662603, −7.50435145912840641727740861664, −6.63598860487121309025505968942, −5.51917294188805204626103501546, −4.70850465692418542576224035578, −3.68230334488970677489848600575, −2.58956281114967806561877743077, −1.13412484114923860162606528911, 1.54877496107114071497128956646, 2.87824183582544406047857827908, 3.91203654212279178657579235975, 4.26867001486833088122555096080, 5.31077224384220333183330845972, 6.87727528050616268498134044990, 7.51768425555050062586204141673, 8.254327050174583316099566318504, 9.424136791910477669956083727616, 10.11414548067440176234721715494

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