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Properties

Label 2-1014-13.5-c2-0-37
Degree $2$
Conductor $1014$
Sign $-0.944 + 0.329i$
Analytic cond. $27.6294$
Root an. cond. $5.25637$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 − i)2-s − 1.73·3-s − 2i·4-s + (−1.48 + 1.48i)5-s + (−1.73 + 1.73i)6-s + (1.99 + 1.99i)7-s + (−2 − 2i)8-s + 2.99·9-s + 2.96i·10-s + (−1.05 − 1.05i)11-s + 3.46i·12-s + 3.98·14-s + (2.56 − 2.56i)15-s − 4·16-s − 21.8i·17-s + (2.99 − 2.99i)18-s + ⋯
L(s)  = 1  + (0.5 − 0.5i)2-s − 0.577·3-s − 0.5i·4-s + (−0.296 + 0.296i)5-s + (−0.288 + 0.288i)6-s + (0.284 + 0.284i)7-s + (−0.250 − 0.250i)8-s + 0.333·9-s + 0.296i·10-s + (−0.0955 − 0.0955i)11-s + 0.288i·12-s + 0.284·14-s + (0.171 − 0.171i)15-s − 0.250·16-s − 1.28i·17-s + (0.166 − 0.166i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.944 + 0.329i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.944 + 0.329i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1014\)    =    \(2 \cdot 3 \cdot 13^{2}\)
Sign: $-0.944 + 0.329i$
Analytic conductor: \(27.6294\)
Root analytic conductor: \(5.25637\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1014} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1014,\ (\ :1),\ -0.944 + 0.329i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.9141071947\)
\(L(\frac12)\) \(\approx\) \(0.9141071947\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 + i)T \)
3 \( 1 + 1.73T \)
13 \( 1 \)
good5 \( 1 + (1.48 - 1.48i)T - 25iT^{2} \)
7 \( 1 + (-1.99 - 1.99i)T + 49iT^{2} \)
11 \( 1 + (1.05 + 1.05i)T + 121iT^{2} \)
17 \( 1 + 21.8iT - 289T^{2} \)
19 \( 1 + (4.68 - 4.68i)T - 361iT^{2} \)
23 \( 1 - 1.76iT - 529T^{2} \)
29 \( 1 - 12.4T + 841T^{2} \)
31 \( 1 + (8.50 - 8.50i)T - 961iT^{2} \)
37 \( 1 + (27.3 + 27.3i)T + 1.36e3iT^{2} \)
41 \( 1 + (-47.8 + 47.8i)T - 1.68e3iT^{2} \)
43 \( 1 + 80.7iT - 1.84e3T^{2} \)
47 \( 1 + (33.7 + 33.7i)T + 2.20e3iT^{2} \)
53 \( 1 + 101.T + 2.80e3T^{2} \)
59 \( 1 + (-12.1 - 12.1i)T + 3.48e3iT^{2} \)
61 \( 1 + 88.5T + 3.72e3T^{2} \)
67 \( 1 + (29.2 - 29.2i)T - 4.48e3iT^{2} \)
71 \( 1 + (9.07 - 9.07i)T - 5.04e3iT^{2} \)
73 \( 1 + (12.1 + 12.1i)T + 5.32e3iT^{2} \)
79 \( 1 + 110.T + 6.24e3T^{2} \)
83 \( 1 + (34.1 - 34.1i)T - 6.88e3iT^{2} \)
89 \( 1 + (-71.5 - 71.5i)T + 7.92e3iT^{2} \)
97 \( 1 + (34.1 - 34.1i)T - 9.40e3iT^{2} \)
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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

−9.532579800815972272734086291968, −8.724432764467427195532737853166, −7.48647025475271052199912829608, −6.82863829987342719107400521679, −5.66425229532557274132848416689, −5.09249294285531008869054612601, −4.03033373092065776625416488233, −3.01258454122548785288495846296, −1.78998113520285015291069413242, −0.26247913673372692687216765018, 1.42703103609862199974296303566, 3.05696072554890304613637002184, 4.39215955083924491106126082527, 4.70042840621610206126082125894, 6.04941446259116658781445562025, 6.44763909645056587814970565652, 7.71626529710220902427505314737, 8.137974777470456968656851186775, 9.227983605056572630526405952882, 10.26499425283038595423666746553

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