Properties

Label 2-2016-224.13-c0-0-0
Degree $2$
Conductor $2016$
Sign $0.980 + 0.195i$
Analytic cond. $1.00611$
Root an. cond. $1.00305$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 + 0.707i)2-s − 1.00i·4-s + (−0.707 + 0.707i)7-s + (0.707 + 0.707i)8-s + (−0.707 − 1.70i)11-s − 1.00i·14-s − 1.00·16-s + (1.70 + 0.707i)22-s + (1 + i)23-s + (0.707 − 0.707i)25-s + (0.707 + 0.707i)28-s + (0.707 − 1.70i)29-s + (0.707 − 0.707i)32-s + (0.707 − 0.292i)37-s + (−0.292 − 0.707i)43-s + (−1.70 + 0.707i)44-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s − 1.00i·4-s + (−0.707 + 0.707i)7-s + (0.707 + 0.707i)8-s + (−0.707 − 1.70i)11-s − 1.00i·14-s − 1.00·16-s + (1.70 + 0.707i)22-s + (1 + i)23-s + (0.707 − 0.707i)25-s + (0.707 + 0.707i)28-s + (0.707 − 1.70i)29-s + (0.707 − 0.707i)32-s + (0.707 − 0.292i)37-s + (−0.292 − 0.707i)43-s + (−1.70 + 0.707i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2016\)    =    \(2^{5} \cdot 3^{2} \cdot 7\)
Sign: $0.980 + 0.195i$
Analytic conductor: \(1.00611\)
Root analytic conductor: \(1.00305\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2016} (685, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2016,\ (\ :0),\ 0.980 + 0.195i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6643277162\)
\(L(\frac12)\) \(\approx\) \(0.6643277162\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 \)
7 \( 1 + (0.707 - 0.707i)T \)
good5 \( 1 + (-0.707 + 0.707i)T^{2} \)
11 \( 1 + (0.707 + 1.70i)T + (-0.707 + 0.707i)T^{2} \)
13 \( 1 + (-0.707 - 0.707i)T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + (-0.707 - 0.707i)T^{2} \)
23 \( 1 + (-1 - i)T + iT^{2} \)
29 \( 1 + (-0.707 + 1.70i)T + (-0.707 - 0.707i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (-0.707 + 0.292i)T + (0.707 - 0.707i)T^{2} \)
41 \( 1 - iT^{2} \)
43 \( 1 + (0.292 + 0.707i)T + (-0.707 + 0.707i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (-0.292 - 0.707i)T + (-0.707 + 0.707i)T^{2} \)
59 \( 1 + (-0.707 + 0.707i)T^{2} \)
61 \( 1 + (0.707 + 0.707i)T^{2} \)
67 \( 1 + (-0.292 + 0.707i)T + (-0.707 - 0.707i)T^{2} \)
71 \( 1 - iT^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 + 1.41iT - T^{2} \)
83 \( 1 + (-0.707 - 0.707i)T^{2} \)
89 \( 1 + iT^{2} \)
97 \( 1 - T^{2} \)
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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

−9.141517351726505674511783627248, −8.527891093878833204801490049250, −7.930835874907794146289414227940, −6.97226978288408891407870975784, −6.04392121211222303345105595965, −5.74180788492969189399364843112, −4.73391224770654405397549027470, −3.29965736067366405116512818444, −2.41743102843893257105563636487, −0.69209541322626166937473202695, 1.20749128807090368073365284731, 2.50140068699164507080516132943, 3.27672315288221537489963663452, 4.40143779719812455335429239030, 5.03732949080677335462113143650, 6.75127767816650704362619139691, 7.01172050250944873800105207393, 7.84440481293590583681858896489, 8.767413032452858623579255992860, 9.500522144434392860901699095966

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