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Properties

Label 2-1014-13.10-c1-0-14
Degree $2$
Conductor $1014$
Sign $0.820 + 0.571i$
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  + (0.866 − 0.5i)2-s + (−0.5 − 0.866i)3-s + (0.499 − 0.866i)4-s + 0.356i·5-s + (−0.866 − 0.499i)6-s + (3.50 + 2.02i)7-s − 0.999i·8-s + (−0.499 + 0.866i)9-s + (0.178 + 0.309i)10-s + (0.789 − 0.455i)11-s − 0.999·12-s + 4.04·14-s + (0.309 − 0.178i)15-s + (−0.5 − 0.866i)16-s + (−1.04 + 1.81i)17-s + 0.999i·18-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (−0.288 − 0.499i)3-s + (0.249 − 0.433i)4-s + 0.159i·5-s + (−0.353 − 0.204i)6-s + (1.32 + 0.765i)7-s − 0.353i·8-s + (−0.166 + 0.288i)9-s + (0.0564 + 0.0977i)10-s + (0.238 − 0.137i)11-s − 0.288·12-s + 1.08·14-s + (0.0798 − 0.0460i)15-s + (−0.125 − 0.216i)16-s + (−0.254 + 0.440i)17-s + 0.235i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.820 + 0.571i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.820 + 0.571i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1014\)    =    \(2 \cdot 3 \cdot 13^{2}\)
Sign: $0.820 + 0.571i$
Analytic conductor: \(8.09683\)
Root analytic conductor: \(2.84549\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1014} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1014,\ (\ :1/2),\ 0.820 + 0.571i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.461688639\)
\(L(\frac12)\) \(\approx\) \(2.461688639\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 \)
good5 \( 1 - 0.356iT - 5T^{2} \)
7 \( 1 + (-3.50 - 2.02i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.789 + 0.455i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.04 - 1.81i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.31 - 2.49i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.24 - 7.35i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.25 + 7.37i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 10.7iT - 31T^{2} \)
37 \( 1 + (-0.533 + 0.307i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (6.58 - 3.80i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.13 - 5.43i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 1.78iT - 47T^{2} \)
53 \( 1 - 10.4T + 53T^{2} \)
59 \( 1 + (5.23 + 3.02i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.55 + 2.69i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-11.7 + 6.78i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (9.94 + 5.74i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 0.533iT - 73T^{2} \)
79 \( 1 + 11.7T + 79T^{2} \)
83 \( 1 + 6.49iT - 83T^{2} \)
89 \( 1 + (-5.62 + 3.24i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.69 - 0.980i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.976325351756940822725559907181, −9.084578782940865117579242103426, −8.032061873789140901405366286523, −7.42503980834440608301548923041, −6.20613076571366442912286945966, −5.51501819178446320920771673807, −4.76755976107501209465511473818, −3.54989075704164476957644783082, −2.27931285492760239260437404053, −1.34523917737577544082594581245, 1.23765240517610202480901393710, 2.93341651136908020287216057062, 4.08587484834577910909500032648, 5.02487849219086721493405995436, 5.20930744792515062734337362571, 6.98412834211701811969964572231, 7.06327529784198616306080909026, 8.514235155156184148131144192477, 8.925929822029413351369363661645, 10.35536360665791827471827517875

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