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Properties

Label 2-2016-224.181-c0-0-0
Degree $2$
Conductor $2016$
Sign $-0.195 - 0.980i$
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 + 0.292i)11-s + 1.00i·14-s − 1.00·16-s + (0.292 + 0.707i)22-s + (1 − i)23-s + (−0.707 − 0.707i)25-s + (−0.707 + 0.707i)28-s + (−0.707 + 0.292i)29-s + (−0.707 − 0.707i)32-s + (−0.707 + 1.70i)37-s + (−1.70 − 0.707i)43-s + (−0.292 + 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 + 0.292i)11-s + 1.00i·14-s − 1.00·16-s + (0.292 + 0.707i)22-s + (1 − i)23-s + (−0.707 − 0.707i)25-s + (−0.707 + 0.707i)28-s + (−0.707 + 0.292i)29-s + (−0.707 − 0.707i)32-s + (−0.707 + 1.70i)37-s + (−1.70 − 0.707i)43-s + (−0.292 + 0.707i)44-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2016\)    =    \(2^{5} \cdot 3^{2} \cdot 7\)
Sign: $-0.195 - 0.980i$
Analytic conductor: \(1.00611\)
Root analytic conductor: \(1.00305\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2016} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2016,\ (\ :0),\ -0.195 - 0.980i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.792214215\)
\(L(\frac12)\) \(\approx\) \(1.792214215\)
\(L(1)\) 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 \( 1 + (-0.707 - 0.292i)T + (0.707 + 0.707i)T^{2} \)
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 \( 1 + (0.707 - 0.292i)T + (0.707 - 0.707i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (0.707 - 1.70i)T + (-0.707 - 0.707i)T^{2} \)
41 \( 1 + iT^{2} \)
43 \( 1 + (1.70 + 0.707i)T + (0.707 + 0.707i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (-1.70 - 0.707i)T + (0.707 + 0.707i)T^{2} \)
59 \( 1 + (0.707 + 0.707i)T^{2} \)
61 \( 1 + (-0.707 + 0.707i)T^{2} \)
67 \( 1 + (-1.70 + 0.707i)T + (0.707 - 0.707i)T^{2} \)
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

−9.240857230605533890157547885781, −8.584981061252636524316655868808, −8.034187419830536117085261529197, −6.99460249377852695177176683309, −6.46465848585577000637703961364, −5.47330184503695290307771097755, −4.84851355508664444668487386147, −4.01394670062281366855847775225, −2.95025200153962225945856468401, −1.86729051005627940736126416218, 1.15600757047971530270336309326, 2.11421982787502270904168949042, 3.54647635587387229175586828304, 3.93632007076975203731266967115, 5.08412844704054861597024899661, 5.61322515139075385520415627401, 6.75967460174509450190532711959, 7.36975742096602303812854120232, 8.469069308929077534750684460364, 9.331781664377649269267356736540

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