Properties

Label 2-2016-7.4-c1-0-6
Degree $2$
Conductor $2016$
Sign $0.895 - 0.444i$
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  + (−1.5 − 2.59i)5-s − 2.64·7-s + (−1.32 + 2.29i)11-s − 4·13-s + (0.5 − 0.866i)17-s + (3.96 + 6.87i)19-s + (−1.32 − 2.29i)23-s + (−2 + 3.46i)25-s + 4·29-s + (1.32 − 2.29i)31-s + (3.96 + 6.87i)35-s + (2.5 + 4.33i)37-s − 8·41-s + 10.5·43-s + (1.32 + 2.29i)47-s + ⋯
L(s)  = 1  + (−0.670 − 1.16i)5-s − 0.999·7-s + (−0.398 + 0.690i)11-s − 1.10·13-s + (0.121 − 0.210i)17-s + (0.910 + 1.57i)19-s + (−0.275 − 0.477i)23-s + (−0.400 + 0.692i)25-s + 0.742·29-s + (0.237 − 0.411i)31-s + (0.670 + 1.16i)35-s + (0.410 + 0.711i)37-s − 1.24·41-s + 1.61·43-s + (0.192 + 0.334i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2016\)    =    \(2^{5} \cdot 3^{2} \cdot 7\)
Sign: $0.895 - 0.444i$
Analytic conductor: \(16.0978\)
Root analytic conductor: \(4.01221\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2016} (865, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2016,\ (\ :1/2),\ 0.895 - 0.444i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9250146640\)
\(L(\frac12)\) \(\approx\) \(0.9250146640\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.64T \)
good5 \( 1 + (1.5 + 2.59i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.32 - 2.29i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 4T + 13T^{2} \)
17 \( 1 + (-0.5 + 0.866i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.96 - 6.87i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.32 + 2.29i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 4T + 29T^{2} \)
31 \( 1 + (-1.32 + 2.29i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.5 - 4.33i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 8T + 41T^{2} \)
43 \( 1 - 10.5T + 43T^{2} \)
47 \( 1 + (-1.32 - 2.29i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.5 + 6.06i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.32 + 2.29i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.5 - 4.33i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.32 - 2.29i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (-4.5 + 7.79i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.32 - 2.29i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 10.5T + 83T^{2} \)
89 \( 1 + (4.5 + 7.79i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 8T + 97T^{2} \)
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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

−9.318944245596956087531850596608, −8.321482675951297588541211606443, −7.74802488653439631891245282891, −6.99465161050351517657755279714, −5.95108204420574895342606301619, −5.07376860927659775474446478432, −4.38412475831685560382062743471, −3.45311620282571182134148518719, −2.33063206810995252516373757938, −0.828907793163875102638107506399, 0.46202895887234430552831230870, 2.62785971823998841128579185016, 3.02439394334540451085005121174, 3.96186882139899164203014525417, 5.11830150325297986493532674103, 6.02336448158187654924155466510, 7.05000896865405432530097841898, 7.20570839079652354057563568316, 8.213858699906801333668902396024, 9.221495686680098656871135458962

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