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Properties

Label 2-1014-13.9-c1-0-3
Degree $2$
Conductor $1014$
Sign $-0.434 - 0.900i$
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 − 0.692·5-s + (0.499 + 0.866i)6-s + (−0.178 − 0.309i)7-s − 0.999·8-s + (−0.499 − 0.866i)9-s + (−0.346 + 0.599i)10-s + (−1.46 + 2.54i)11-s + 0.999·12-s − 0.356·14-s + (0.346 − 0.599i)15-s + (−0.5 + 0.866i)16-s + (−3.35 − 5.81i)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 − 0.309·5-s + (0.204 + 0.353i)6-s + (−0.0674 − 0.116i)7-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.109 + 0.189i)10-s + (−0.443 + 0.767i)11-s + 0.288·12-s − 0.0953·14-s + (0.0893 − 0.154i)15-s + (−0.125 + 0.216i)16-s + (−0.814 − 1.41i)17-s − 0.235·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.434 - 0.900i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.434 - 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1014\)    =    \(2 \cdot 3 \cdot 13^{2}\)
Sign: $-0.434 - 0.900i$
Analytic conductor: \(8.09683\)
Root analytic conductor: \(2.84549\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1014} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1014,\ (\ :1/2),\ -0.434 - 0.900i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4981036119\)
\(L(\frac12)\) \(\approx\) \(0.4981036119\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.692T + 5T^{2} \)
7 \( 1 + (0.178 + 0.309i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.46 - 2.54i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (3.35 + 5.81i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.60 - 6.24i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.19 - 2.07i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.91 - 6.78i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 2.76T + 31T^{2} \)
37 \( 1 + (5.04 - 8.74i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.44 - 4.23i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.29 + 5.70i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 4.98T + 47T^{2} \)
53 \( 1 + 8.88T + 53T^{2} \)
59 \( 1 + (0.821 + 1.42i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.24 - 5.62i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.76 - 11.7i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.40 + 5.89i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 3.18T + 73T^{2} \)
79 \( 1 - 15.0T + 79T^{2} \)
83 \( 1 + 14.8T + 83T^{2} \)
89 \( 1 + (0.198 - 0.343i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.208 - 0.361i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.15405256642020523010776795448, −9.728358933508565583141735644948, −8.805117237628490021889434122545, −7.65745669654433669221173038367, −6.86181132571435632501989369957, −5.56795876770478223448103426704, −4.96910261135446525418110451165, −3.96269573190340499923410104358, −3.11270572757087966470961597376, −1.72034642022907711041074621378, 0.19975918286848357253304924341, 2.17928928253061191974267858383, 3.48827745238008113717873982692, 4.49677578527614485946406127759, 5.61393219746698411816654854425, 6.17860352714225922035976414004, 7.15702166001371100779833838455, 7.88294115316510706692999630130, 8.633681983481930848904687258140, 9.481769914556751640028525255568

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