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Properties

Label 2-1014-13.3-c1-0-21
Degree $2$
Conductor $1014$
Sign $0.309 + 0.950i$
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  + (0.5 + 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s − 0.356·5-s + (0.499 − 0.866i)6-s + (2.02 − 3.50i)7-s − 0.999·8-s + (−0.499 + 0.866i)9-s + (−0.178 − 0.309i)10-s + (−0.455 − 0.789i)11-s + 0.999·12-s + 4.04·14-s + (0.178 + 0.309i)15-s + (−0.5 − 0.866i)16-s + (1.04 − 1.81i)17-s − 0.999·18-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s − 0.159·5-s + (0.204 − 0.353i)6-s + (0.765 − 1.32i)7-s − 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.0564 − 0.0977i)10-s + (−0.137 − 0.238i)11-s + 0.288·12-s + 1.08·14-s + (0.0460 + 0.0798i)15-s + (−0.125 − 0.216i)16-s + (0.254 − 0.440i)17-s − 0.235·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.309 + 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.309 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1014\)    =    \(2 \cdot 3 \cdot 13^{2}\)
Sign: $0.309 + 0.950i$
Analytic conductor: \(8.09683\)
Root analytic conductor: \(2.84549\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1014} (991, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1014,\ (\ :1/2),\ 0.309 + 0.950i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.365142003\)
\(L(\frac12)\) \(\approx\) \(1.365142003\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 \)
good5 \( 1 + 0.356T + 5T^{2} \)
7 \( 1 + (-2.02 + 3.50i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.455 + 0.789i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.04 + 1.81i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.49 - 4.31i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.24 + 7.35i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.25 + 7.37i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 10.7T + 31T^{2} \)
37 \( 1 + (0.307 + 0.533i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.80 + 6.58i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.13 + 5.43i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 1.78T + 47T^{2} \)
53 \( 1 - 10.4T + 53T^{2} \)
59 \( 1 + (3.02 - 5.23i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.55 + 2.69i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.78 - 11.7i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-5.74 + 9.94i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 0.533T + 73T^{2} \)
79 \( 1 + 11.7T + 79T^{2} \)
83 \( 1 - 6.49T + 83T^{2} \)
89 \( 1 + (3.24 + 5.62i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.980 - 1.69i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.02831675984439679603788717991, −8.487331754253974679301220059353, −7.967025791583270326494981050996, −7.31336053145218824467417264666, −6.41049398074377675453680147042, −5.61778792038084196273398502997, −4.43655928809738288518411403823, −3.89365918929119662696642739293, −2.20709481537079122906486386861, −0.57600608628593051159801864688, 1.67746266576274217185598461638, 2.76886102301300058908805940575, 3.93609044471581697973583267465, 4.92703687372748361147181213212, 5.53318501637080290256711822110, 6.43216645871680761827437317658, 7.84815464519685452518539074740, 8.612030770339504109697112897958, 9.462534824421230683092812138532, 10.13324040601394647681699961941

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