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Properties

Label 2-2016-7.4-c1-0-15
Degree $2$
Conductor $2016$
Sign $0.701 - 0.712i$
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  + (0.5 + 0.866i)5-s + (2.5 − 0.866i)7-s + (0.5 − 0.866i)11-s + (−4 + 6.92i)17-s + (2 + 3.46i)19-s + (2 + 3.46i)23-s + (2 − 3.46i)25-s + 5·29-s + (−3.5 + 6.06i)31-s + (2 + 1.73i)35-s + (−4 − 6.92i)37-s − 4·41-s + 10·43-s + (3 + 5.19i)47-s + (5.5 − 4.33i)49-s + ⋯
L(s)  = 1  + (0.223 + 0.387i)5-s + (0.944 − 0.327i)7-s + (0.150 − 0.261i)11-s + (−0.970 + 1.68i)17-s + (0.458 + 0.794i)19-s + (0.417 + 0.722i)23-s + (0.400 − 0.692i)25-s + 0.928·29-s + (−0.628 + 1.08i)31-s + (0.338 + 0.292i)35-s + (−0.657 − 1.13i)37-s − 0.624·41-s + 1.52·43-s + (0.437 + 0.757i)47-s + (0.785 − 0.618i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.701 - 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.701 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2016\)    =    \(2^{5} \cdot 3^{2} \cdot 7\)
Sign: $0.701 - 0.712i$
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: \((2,\ 2016,\ (\ :1/2),\ 0.701 - 0.712i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.007746072\)
\(L(\frac12)\) \(\approx\) \(2.007746072\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.5 + 0.866i)T \)
good5 \( 1 + (-0.5 - 0.866i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + (4 - 6.92i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2 - 3.46i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2 - 3.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 5T + 29T^{2} \)
31 \( 1 + (3.5 - 6.06i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4 + 6.92i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 4T + 41T^{2} \)
43 \( 1 - 10T + 43T^{2} \)
47 \( 1 + (-3 - 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.5 + 7.79i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1 - 1.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1 - 1.73i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + (1 - 1.73i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.5 - 7.79i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 3T + 83T^{2} \)
89 \( 1 + (3 + 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + T + 97T^{2} \)
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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

−9.068327762958757799580664202907, −8.495083011601441179985838463628, −7.72739416295976960646275621705, −6.90347428186296502232815215224, −6.09680596053847495464677489411, −5.27721344113607903258317983061, −4.28485181123165383435234236296, −3.52048864702580338578914134711, −2.22669533288683847372105892941, −1.27287052238439396897820266746, 0.803913703587507309127474327998, 2.10157404764635434982434892135, 2.96558661939030995586903722190, 4.48542001795131037799890984259, 4.88082321367443428607565594741, 5.70365545540187800779698117644, 6.89784169544246728066116567569, 7.35973366164228440268427273115, 8.478657507005296287502795254695, 8.994425015705436011581647918823

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