Properties

Degree $2$
Conductor $2016$
Sign $-0.991 - 0.126i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2 + 3.46i)5-s + (−2.5 + 0.866i)7-s + (3 + 5.19i)11-s + 5·13-s + (1 + 1.73i)17-s + (−0.5 + 0.866i)19-s + (−3 + 5.19i)23-s + (−5.49 − 9.52i)25-s + (1.5 + 2.59i)31-s + (2.00 − 10.3i)35-s + (−1.5 + 2.59i)37-s + 6·41-s + 5·43-s + (−2 + 3.46i)47-s + (5.5 − 4.33i)49-s + ⋯
L(s)  = 1  + (−0.894 + 1.54i)5-s + (−0.944 + 0.327i)7-s + (0.904 + 1.56i)11-s + 1.38·13-s + (0.242 + 0.420i)17-s + (−0.114 + 0.198i)19-s + (−0.625 + 1.08i)23-s + (−1.09 − 1.90i)25-s + (0.269 + 0.466i)31-s + (0.338 − 1.75i)35-s + (−0.246 + 0.427i)37-s + 0.937·41-s + 0.762·43-s + (−0.291 + 0.505i)47-s + (0.785 − 0.618i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2016\)    =    \(2^{5} \cdot 3^{2} \cdot 7\)
Sign: $-0.991 - 0.126i$
Motivic weight: \(1\)
Character: $\chi_{2016} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2016,\ (\ :1/2),\ -0.991 - 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.083012667\)
\(L(\frac12)\) \(\approx\) \(1.083012667\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.5 - 0.866i)T \)
good5 \( 1 + (2 - 3.46i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-3 - 5.19i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 5T + 13T^{2} \)
17 \( 1 + (-1 - 1.73i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3 - 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + (-1.5 - 2.59i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.5 - 2.59i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 - 5T + 43T^{2} \)
47 \( 1 + (2 - 3.46i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3 + 5.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3 + 5.19i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1 + 1.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.5 + 6.06i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 16T + 71T^{2} \)
73 \( 1 + (-1.5 - 2.59i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.5 - 9.52i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 12T + 83T^{2} \)
89 \( 1 + (-2 + 3.46i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 6T + 97T^{2} \)
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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

−9.705669078799358469279935259753, −8.748867424409151745968061019412, −7.76732161916545072025508817617, −7.12186113737288095602389529301, −6.44986139641714886459493366663, −5.91783845579342137278276328283, −4.29361359603048031755335932096, −3.68102080081483078686252494703, −2.98474324475393554791208926417, −1.71464561998985268449813331184, 0.45630287248820541482970382664, 1.15789215053590767730341613715, 3.09834762131747131662728026251, 3.93509239316057844557849880793, 4.40511545942400333726023419678, 5.81357087601946103823510898100, 6.14073094259087855165751058282, 7.32443263578210754350423006043, 8.228226317465367332758961560617, 8.874871154837142022026726549725

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