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Properties

Label 2-2016-7.2-c1-0-16
Degree $2$
Conductor $2016$
Sign $-0.285 - 0.958i$
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  + (−1.46 + 2.52i)5-s + (1.80 + 1.93i)7-s + (2.26 + 3.92i)11-s + 5.53·13-s + (2.23 + 3.87i)17-s + (−0.342 + 0.593i)19-s + (1 − 1.73i)23-s + (−1.76 − 3.05i)25-s − 4.29·29-s + (−4.03 − 6.99i)31-s + (−7.53 + 1.73i)35-s + (2.76 − 4.79i)37-s − 2.73·41-s + 3.78·43-s + (5.53 − 9.58i)47-s + ⋯
L(s)  = 1  + (−0.653 + 1.13i)5-s + (0.681 + 0.731i)7-s + (0.683 + 1.18i)11-s + 1.53·13-s + (0.542 + 0.939i)17-s + (−0.0785 + 0.136i)19-s + (0.208 − 0.361i)23-s + (−0.353 − 0.611i)25-s − 0.796·29-s + (−0.725 − 1.25i)31-s + (−1.27 + 0.292i)35-s + (0.454 − 0.787i)37-s − 0.427·41-s + 0.577·43-s + (0.806 − 1.39i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.285 - 0.958i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.285 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2016\)    =    \(2^{5} \cdot 3^{2} \cdot 7\)
Sign: $-0.285 - 0.958i$
Analytic conductor: \(16.0978\)
Root analytic conductor: \(4.01221\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2016} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2016,\ (\ :1/2),\ -0.285 - 0.958i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.836553165\)
\(L(\frac12)\) \(\approx\) \(1.836553165\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.80 - 1.93i)T \)
good5 \( 1 + (1.46 - 2.52i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.26 - 3.92i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 5.53T + 13T^{2} \)
17 \( 1 + (-2.23 - 3.87i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.342 - 0.593i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1 + 1.73i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 4.29T + 29T^{2} \)
31 \( 1 + (4.03 + 6.99i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.76 + 4.79i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 2.73T + 41T^{2} \)
43 \( 1 - 3.78T + 43T^{2} \)
47 \( 1 + (-5.53 + 9.58i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.93 - 10.2i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.734 - 1.27i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.53 - 9.58i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.18 - 10.7i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 7.06T + 71T^{2} \)
73 \( 1 + (2.76 + 4.79i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-8.32 + 14.4i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 1.46T + 83T^{2} \)
89 \( 1 + (8.76 - 15.1i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 1.46T + 97T^{2} \)
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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

−9.223647896493441174153022168471, −8.624486645440401963806766815404, −7.68723328548389005494237410587, −7.18560283199015247750357565687, −6.17263194033582597769872811463, −5.60592071757188955481800294222, −4.15273916280828163900121443181, −3.78620366840513762759260956315, −2.51086000549811093128638824023, −1.51993012641380042220178922597, 0.77158927855791858219528555755, 1.39849261316835131820922807211, 3.34862870558694418092932405035, 3.90983870426284432254076023925, 4.84379771969513017459645190831, 5.58677980965941419725211678761, 6.57496542789651765640012562665, 7.54839193802303781860229560758, 8.264788773904653868128039686329, 8.772537886157675743414359453889

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