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Properties

Label 2-1014-13.3-c1-0-11
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

\(L(1)\) \(\approx\) \(1.317393546\)
\(L(\frac12)\) \(\approx\) \(1.317393546\)
\(L(\frac{3}{2})\) 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

−9.916990580057499581559890464604, −9.099885430832980605017371436170, −8.524952657966552030407085928034, −7.31934016651896655492546429919, −6.51002941598703281298339055995, −5.62213911526220617674109292968, −4.73882444032061030566361779909, −3.14297483034777175511540972106, −2.32694789004965129489021697840, −1.11010681609343546837981652507, 0.891627287176597631210563176832, 2.59038320550320366500691043304, 4.02858383587444260502653930298, 4.90839654789549741958747741100, 6.01155636052248047055835378070, 6.39493597316988838302423067692, 7.52603363573901195948023823579, 8.360181206716990615801872853068, 9.443730558809914709056438519300, 9.943605321589757556460227478288

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