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Properties

Label 2-1014-13.9-c1-0-22
Degree $2$
Conductor $1014$
Sign $-0.755 + 0.655i$
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 + 1.73·5-s + (−0.499 − 0.866i)6-s + (−0.633 − 1.09i)7-s − 0.999·8-s + (−0.499 − 0.866i)9-s + (0.866 − 1.49i)10-s + (−0.633 + 1.09i)11-s − 0.999·12-s − 1.26·14-s + (0.866 − 1.49i)15-s + (−0.5 + 0.866i)16-s + (−2.59 − 4.5i)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 + 0.774·5-s + (−0.204 − 0.353i)6-s + (−0.239 − 0.415i)7-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (0.273 − 0.474i)10-s + (−0.191 + 0.331i)11-s − 0.288·12-s − 0.338·14-s + (0.223 − 0.387i)15-s + (−0.125 + 0.216i)16-s + (−0.630 − 1.09i)17-s − 0.235·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 + 0.655i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.755 + 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1014\)    =    \(2 \cdot 3 \cdot 13^{2}\)
Sign: $-0.755 + 0.655i$
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: \((2,\ 1014,\ (\ :1/2),\ -0.755 + 0.655i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.012088159\)
\(L(\frac12)\) \(\approx\) \(2.012088159\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.73T + 5T^{2} \)
7 \( 1 + (0.633 + 1.09i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.633 - 1.09i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (2.59 + 4.5i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.36 + 4.09i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.09 + 7.09i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.5 + 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 9.46T + 31T^{2} \)
37 \( 1 + (1.5 - 2.59i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.23 - 5.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.09 - 3.63i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 4.73T + 47T^{2} \)
53 \( 1 - 3T + 53T^{2} \)
59 \( 1 + (-6.92 - 12i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.59 + 13.1i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.63 - 6.29i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.09 - 1.90i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 12.1T + 73T^{2} \)
79 \( 1 - 8.39T + 79T^{2} \)
83 \( 1 - 5.66T + 83T^{2} \)
89 \( 1 + (4.73 - 8.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3 + 5.19i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.738952923155657607792244601942, −8.978636071378286550807172742507, −8.079428425812033542607047209843, −6.77093812575731718446507195545, −6.46063652865219640565881988374, −5.07542507110338206978610587255, −4.37230331736946274036767094605, −2.87091240016748578150206092308, −2.28671665363405871728820345908, −0.77903028533538935146245994635, 1.94108609396576904762495473164, 3.19013151598015824790781444438, 4.12825681442363055605613855590, 5.28060887590775151590693126117, 5.91227576191916703090274079315, 6.69147128074287386935343135040, 7.891794886274741795671852665883, 8.627687026450717166823419758741, 9.332406213989283392342809325266, 10.15989388368348579484825147340

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