Properties

Label 2-1014-13.3-c1-0-1
Degree $2$
Conductor $1014$
Sign $-0.859 - 0.511i$
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 − 2·5-s + (0.499 − 0.866i)6-s + (1 − 1.73i)7-s − 0.999·8-s + (−0.499 + 0.866i)9-s + (−1 − 1.73i)10-s + 0.999·12-s + 1.99·14-s + (1 + 1.73i)15-s + (−0.5 − 0.866i)16-s + (−1 + 1.73i)17-s − 0.999·18-s + (−3 + 5.19i)19-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s − 0.894·5-s + (0.204 − 0.353i)6-s + (0.377 − 0.654i)7-s − 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.316 − 0.547i)10-s + 0.288·12-s + 0.534·14-s + (0.258 + 0.447i)15-s + (−0.125 − 0.216i)16-s + (−0.242 + 0.420i)17-s − 0.235·18-s + (−0.688 + 1.19i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.859 - 0.511i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1014\)    =    \(2 \cdot 3 \cdot 13^{2}\)
Sign: $-0.859 - 0.511i$
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: \((2,\ 1014,\ (\ :1/2),\ -0.859 - 0.511i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6328553890\)
\(L(\frac12)\) \(\approx\) \(0.6328553890\)
\(L(\frac{3}{2})\) 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 \( 1 + 2T + 5T^{2} \)
7 \( 1 + (-1 + 1.73i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (1 - 1.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3 - 5.19i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2 - 3.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-5 - 8.66i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 10T + 31T^{2} \)
37 \( 1 + (4 + 6.92i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-5 - 8.66i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2 + 3.46i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 12T + 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + (2 - 3.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1 - 1.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1 + 1.73i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 4T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 4T + 83T^{2} \)
89 \( 1 + (-3 - 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (6 - 10.3i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.59444209236957201935860917780, −9.242853529916320486687806449514, −8.281947563588531317949758801687, −7.66733790804310313998396314128, −7.05838680628993442223436708799, −6.12310376046274966148055926048, −5.15324314649396222373797726002, −4.16881538887795553586051332917, −3.39242260177997362426363694896, −1.57575425017320650177578496580, 0.26578044992040045765083966164, 2.20574296222107358145287473770, 3.29504093913846994856809455123, 4.38822627681498898089242107128, 4.91063109710411538477994255682, 5.99900207699623952960150162225, 7.00149520428536798882654228377, 8.171270459963476089555286986800, 8.885104135941101310112792237722, 9.680266144699127355721488924751

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