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Properties

Label 2-2016-56.37-c1-0-6
Degree $2$
Conductor $2016$
Sign $0.249 - 0.968i$
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  + (−1.98 − 1.14i)5-s + (1.05 + 2.42i)7-s + (3.36 − 1.94i)11-s + 3.33i·13-s + (0.143 + 0.248i)17-s + (−2.41 − 1.39i)19-s + (−3.26 + 5.65i)23-s + (0.132 + 0.229i)25-s − 5.53i·29-s + (3.72 + 6.44i)31-s + (0.684 − 6.03i)35-s + (−5.15 − 2.97i)37-s + 3.51·41-s + 11.2i·43-s + (−0.0435 + 0.0753i)47-s + ⋯
L(s)  = 1  + (−0.888 − 0.513i)5-s + (0.399 + 0.916i)7-s + (1.01 − 0.585i)11-s + 0.924i·13-s + (0.0348 + 0.0603i)17-s + (−0.553 − 0.319i)19-s + (−0.681 + 1.17i)23-s + (0.0265 + 0.0459i)25-s − 1.02i·29-s + (0.668 + 1.15i)31-s + (0.115 − 1.01i)35-s + (−0.847 − 0.489i)37-s + 0.549·41-s + 1.71i·43-s + (−0.00634 + 0.0109i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.249 - 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.249 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2016\)    =    \(2^{5} \cdot 3^{2} \cdot 7\)
Sign: $0.249 - 0.968i$
Analytic conductor: \(16.0978\)
Root analytic conductor: \(4.01221\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2016} (1297, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2016,\ (\ :1/2),\ 0.249 - 0.968i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.265885929\)
\(L(\frac12)\) \(\approx\) \(1.265885929\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.05 - 2.42i)T \)
good5 \( 1 + (1.98 + 1.14i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-3.36 + 1.94i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 3.33iT - 13T^{2} \)
17 \( 1 + (-0.143 - 0.248i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.41 + 1.39i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.26 - 5.65i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 5.53iT - 29T^{2} \)
31 \( 1 + (-3.72 - 6.44i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.15 + 2.97i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 3.51T + 41T^{2} \)
43 \( 1 - 11.2iT - 43T^{2} \)
47 \( 1 + (0.0435 - 0.0753i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.11 + 3.52i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.76 - 2.17i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.20 - 3.58i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-11.2 + 6.51i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 6.18T + 71T^{2} \)
73 \( 1 + (-6.93 - 12.0i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.49 - 7.79i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 17.6iT - 83T^{2} \)
89 \( 1 + (8.59 - 14.8i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 6.46T + 97T^{2} \)
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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

−9.180942602151750218014161574690, −8.455417427413407321195853142830, −8.047974729162207124406687574592, −6.91666141276085887136827553908, −6.17320089707176136304231873960, −5.26804573646279007524616449621, −4.30431167553405272204026809698, −3.72544722128777305718907154579, −2.39090754227849804580591932239, −1.21628276344347543894696023923, 0.50644242609538686433362103519, 1.92910518436731655474874149213, 3.30789156595552113746421595776, 4.02816267087850774593030575579, 4.67339051501010636505537032913, 5.91152491744437350105304157805, 6.86608328233631409986248997240, 7.36003515757297949690245172348, 8.135654200346407058011343257447, 8.803955549106158534372107711257

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