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Properties

Label 2-2016-672.251-c0-0-3
Degree $2$
Conductor $2016$
Sign $0.358 + 0.933i$
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  − 2-s + 4-s + (0.707 − 0.707i)7-s − 8-s + (0.707 − 1.70i)11-s + (−0.707 + 0.707i)14-s + 16-s + (−0.707 + 1.70i)22-s + (−0.707 − 0.707i)25-s + (0.707 − 0.707i)28-s + (−0.707 + 0.292i)29-s − 32-s + (−1.70 − 0.707i)37-s + (0.707 + 0.292i)43-s + (0.707 − 1.70i)44-s + ⋯
L(s)  = 1  − 2-s + 4-s + (0.707 − 0.707i)7-s − 8-s + (0.707 − 1.70i)11-s + (−0.707 + 0.707i)14-s + 16-s + (−0.707 + 1.70i)22-s + (−0.707 − 0.707i)25-s + (0.707 − 0.707i)28-s + (−0.707 + 0.292i)29-s − 32-s + (−1.70 − 0.707i)37-s + (0.707 + 0.292i)43-s + (0.707 − 1.70i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.358 + 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.358 + 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2016\)    =    \(2^{5} \cdot 3^{2} \cdot 7\)
Sign: $0.358 + 0.933i$
Analytic conductor: \(1.00611\)
Root analytic conductor: \(1.00305\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2016} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2016,\ (\ :0),\ 0.358 + 0.933i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7884731239\)
\(L(\frac12)\) \(\approx\) \(0.7884731239\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
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 \( 1 - iT^{2} \)
29 \( 1 + (0.707 - 0.292i)T + (0.707 - 0.707i)T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (1.70 + 0.707i)T + (0.707 + 0.707i)T^{2} \)
41 \( 1 - iT^{2} \)
43 \( 1 + (-0.707 - 0.292i)T + (0.707 + 0.707i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (-0.707 - 0.292i)T + (0.707 + 0.707i)T^{2} \)
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 \( 1 + (-1 - i)T + iT^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 - 1.41T + T^{2} \)
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

−8.945723260950830811668916316749, −8.563479323621168470184750676157, −7.74481610467730831757484507140, −7.05142152633965810154422445627, −6.15169297187923427910019536140, −5.45535340260555968904411124971, −4.05194059186810699415429646301, −3.26683398549657741636414371551, −1.93449400373786570754117932742, −0.813611574727140676023112931511, 1.63816856164113404029374405661, 2.17722260769148271210566573378, 3.56070618477746718440297683794, 4.72904336615261449074694070346, 5.61879126760612385310179043535, 6.57757259852417036526733768489, 7.34938363666297140220217263054, 7.905840252877792717791139820249, 8.910757167825380434902953708351, 9.326017429491530066617888854318

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