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Properties

Label 2-2016-7.4-c1-0-11
Degree $2$
Conductor $2016$
Sign $0.991 - 0.126i$
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  + (−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: $0.991 - 0.126i$
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: \((2,\ 2016,\ (\ :1/2),\ 0.991 - 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.601087668\)
\(L(\frac12)\) \(\approx\) \(1.601087668\)
\(L(\frac{3}{2})\) 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

−9.100487577212695413056994277020, −8.242218281435183352536760716263, −7.85330466796297915004207747358, −7.10239417724314872657937766386, −5.61524747117450991024449085357, −5.05306819502997888894083376791, −4.45467933550838778263732910673, −3.52980311674354794443760840815, −1.95279464218411590062008928367, −1.04448515017775994013852254602, 0.73383834323773110172313976429, 2.42731402077276535099807093052, 3.38233598413454419126362916744, 3.89955087739940349222494274180, 5.14721778978883171253185514551, 6.13378661811690869059310885874, 6.75483897660005135556884315620, 7.77478051231418797909575514521, 8.160177517000363723459361580824, 8.824755302780132110514277554088

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