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Properties

Label 2-2016-7.4-c1-0-33
Degree $2$
Conductor $2016$
Sign $-0.480 + 0.877i$
Analytic cond. $16.0978$
Root an. cond. $4.01221$
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  + (0.227 + 0.393i)5-s + (2.16 − 1.51i)7-s + (−2.89 + 5.01i)11-s − 5.88·13-s + (1.45 − 2.51i)17-s + (−2.94 − 5.09i)19-s + (−1.45 − 2.51i)23-s + (2.39 − 4.15i)25-s − 3.54·29-s + (2.16 − 3.75i)31-s + (1.09 + 0.509i)35-s + (−3.85 − 6.67i)37-s − 9.58·41-s + 10.7·43-s + (2.45 + 4.25i)47-s + ⋯
L(s)  = 1  + (0.101 + 0.176i)5-s + (0.819 − 0.572i)7-s + (−0.873 + 1.51i)11-s − 1.63·13-s + (0.352 − 0.611i)17-s + (−0.674 − 1.16i)19-s + (−0.303 − 0.525i)23-s + (0.479 − 0.830i)25-s − 0.658·29-s + (0.389 − 0.674i)31-s + (0.184 + 0.0861i)35-s + (−0.633 − 1.09i)37-s − 1.49·41-s + 1.64·43-s + (0.358 + 0.620i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.480 + 0.877i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.480 + 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2016\)    =    \(2^{5} \cdot 3^{2} \cdot 7\)
Sign: $-0.480 + 0.877i$
Analytic conductor: \(16.0978\)
Root analytic conductor: \(4.01221\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2016} (865, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2016,\ (\ :1/2),\ -0.480 + 0.877i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8723088208\)
\(L(\frac12)\) \(\approx\) \(0.8723088208\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + (-2.16 + 1.51i)T \)
good5 \( 1 + (-0.227 - 0.393i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.89 - 5.01i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 5.88T + 13T^{2} \)
17 \( 1 + (-1.45 + 2.51i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.94 + 5.09i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.45 + 2.51i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 3.54T + 29T^{2} \)
31 \( 1 + (-2.16 + 3.75i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.85 + 6.67i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 9.58T + 41T^{2} \)
43 \( 1 - 10.7T + 43T^{2} \)
47 \( 1 + (-2.45 - 4.25i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.56 + 11.3i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.896 - 1.55i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.33 + 4.04i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.94 - 6.82i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 0.909T + 71T^{2} \)
73 \( 1 + (2.60 - 4.50i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.37 - 2.38i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 9.97T + 83T^{2} \)
89 \( 1 + (2.45 + 4.25i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 5.79T + 97T^{2} \)
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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

−8.911669637172379288922221958656, −7.925264683441895240378617611150, −7.24749521408951747105389085242, −6.92454388467600760111911589443, −5.45200768671608656978697617021, −4.74974524893384845669926920839, −4.27176056514759985365322239329, −2.59268135541074536305180937797, −2.11486516122628116274736297943, −0.29325182691618461834602913760, 1.47929899361279732280347728617, 2.55988149442604569664450496440, 3.51274598499721509137705356276, 4.73358691473855304666908508792, 5.47867778122576346531691515360, 5.95064821429404419277484015549, 7.25118384494948000312997672334, 7.975460364554741597927072903291, 8.520546670234582569906910828447, 9.273434531565755583096970422864

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