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Properties

Label 2-2016-504.499-c0-0-1
Degree $2$
Conductor $2016$
Sign $0.971 - 0.235i$
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  + 3-s + (0.866 − 0.5i)5-s + i·7-s + 9-s + (−0.5 + 0.866i)11-s + (−0.866 − 0.5i)13-s + (0.866 − 0.5i)15-s + (0.5 + 0.866i)17-s + (0.5 − 0.866i)19-s + i·21-s + (−0.866 + 0.5i)23-s + 27-s + (0.866 − 0.5i)29-s + (−0.5 + 0.866i)33-s + (0.5 + 0.866i)35-s + ⋯
L(s)  = 1  + 3-s + (0.866 − 0.5i)5-s + i·7-s + 9-s + (−0.5 + 0.866i)11-s + (−0.866 − 0.5i)13-s + (0.866 − 0.5i)15-s + (0.5 + 0.866i)17-s + (0.5 − 0.866i)19-s + i·21-s + (−0.866 + 0.5i)23-s + 27-s + (0.866 − 0.5i)29-s + (−0.5 + 0.866i)33-s + (0.5 + 0.866i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 - 0.235i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 - 0.235i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2016\)    =    \(2^{5} \cdot 3^{2} \cdot 7\)
Sign: $0.971 - 0.235i$
Analytic conductor: \(1.00611\)
Root analytic conductor: \(1.00305\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2016} (751, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2016,\ (\ :0),\ 0.971 - 0.235i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.837293839\)
\(L(\frac12)\) \(\approx\) \(1.837293839\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
7 \( 1 - iT \)
good5 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
41 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + 2iT - T^{2} \)
83 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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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

−9.288610713980954208986416134120, −8.750519803964511533521267146529, −7.85642503880076709538806310171, −7.27837812691924048384438245103, −6.09101298662472228554498440395, −5.30210141276762484865119996928, −4.62218694574782641427657427006, −3.33093884593254573198025812897, −2.36036768034826701041316288468, −1.76525679679166117442077359939, 1.41357876824931239496654712700, 2.60162087377778855620030738079, 3.25249313304339734490980712099, 4.30597100979190554303236632324, 5.23396914768327144724107978224, 6.36585551168249298336313573523, 7.01367441530388922669129005379, 7.84958737970590527664513956998, 8.379370142379055329173638219724, 9.574122687552991383256865577611

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