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Properties

Label 2-1014-13.9-c1-0-7
Degree $2$
Conductor $1014$
Sign $-0.0128 - 0.999i$
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 − 3·5-s + (−0.499 − 0.866i)6-s + (1 + 1.73i)7-s + 0.999·8-s + (−0.499 − 0.866i)9-s + (1.5 − 2.59i)10-s + (3 − 5.19i)11-s + 0.999·12-s − 1.99·14-s + (1.5 − 2.59i)15-s + (−0.5 + 0.866i)16-s + (1.5 + 2.59i)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 − 1.34·5-s + (−0.204 − 0.353i)6-s + (0.377 + 0.654i)7-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (0.474 − 0.821i)10-s + (0.904 − 1.56i)11-s + 0.288·12-s − 0.534·14-s + (0.387 − 0.670i)15-s + (−0.125 + 0.216i)16-s + (0.363 + 0.630i)17-s + 0.235·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0128 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0128 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1014\)    =    \(2 \cdot 3 \cdot 13^{2}\)
Sign: $-0.0128 - 0.999i$
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: \((2,\ 1014,\ (\ :1/2),\ -0.0128 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9058087635\)
\(L(\frac12)\) \(\approx\) \(0.9058087635\)
\(L(\frac{3}{2})\) 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 \( 1 + 3T + 5T^{2} \)
7 \( 1 + (-1 - 1.73i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-3 + 5.19i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1 - 1.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.5 - 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + (3.5 - 6.06i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.5 - 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5 - 8.66i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 6T + 47T^{2} \)
53 \( 1 - 3T + 53T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.5 - 6.06i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5 - 8.66i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3 - 5.19i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 13T + 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 - 6T + 83T^{2} \)
89 \( 1 + (-9 + 15.5i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-7 - 12.1i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.17098017482769254764825271993, −9.001977840954471147593601840267, −8.466335345396784134129871665618, −7.943477552843235372051692402466, −6.71231790152800578131661867995, −5.98771638121019862726517837465, −5.01211841824957368727230080658, −4.03299847662459628965932896770, −3.16747251500587658017099786223, −0.965010919160016085248344419349, 0.68828884815438676489826169522, 1.94859706041883630553691688647, 3.48852194824463752075217887845, 4.27192319431339598584312167493, 5.11876807590698223064728580114, 6.81881506682350185880462802372, 7.40517446390885702761783735686, 7.83575555622262577170252828989, 9.020612880436558334322838123225, 9.749759990523999846255781611579

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