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Properties

Label 2-2016-7.4-c1-0-16
Degree $2$
Conductor $2016$
Sign $0.285 - 0.958i$
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  + (1.46 + 2.52i)5-s + (−1.80 + 1.93i)7-s + (2.26 − 3.92i)11-s + 5.53·13-s + (−2.23 + 3.87i)17-s + (0.342 + 0.593i)19-s + (1 + 1.73i)23-s + (−1.76 + 3.05i)25-s + 4.29·29-s + (4.03 − 6.99i)31-s + (−7.53 − 1.73i)35-s + (2.76 + 4.79i)37-s + 2.73·41-s − 3.78·43-s + (5.53 + 9.58i)47-s + ⋯
L(s)  = 1  + (0.653 + 1.13i)5-s + (−0.681 + 0.731i)7-s + (0.683 − 1.18i)11-s + 1.53·13-s + (−0.542 + 0.939i)17-s + (0.0785 + 0.136i)19-s + (0.208 + 0.361i)23-s + (−0.353 + 0.611i)25-s + 0.796·29-s + (0.725 − 1.25i)31-s + (−1.27 − 0.292i)35-s + (0.454 + 0.787i)37-s + 0.427·41-s − 0.577·43-s + (0.806 + 1.39i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.285 - 0.958i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.285 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2016\)    =    \(2^{5} \cdot 3^{2} \cdot 7\)
Sign: $0.285 - 0.958i$
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: \((2,\ 2016,\ (\ :1/2),\ 0.285 - 0.958i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.006309189\)
\(L(\frac12)\) \(\approx\) \(2.006309189\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.80 - 1.93i)T \)
good5 \( 1 + (-1.46 - 2.52i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.26 + 3.92i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 5.53T + 13T^{2} \)
17 \( 1 + (2.23 - 3.87i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.342 - 0.593i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1 - 1.73i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 4.29T + 29T^{2} \)
31 \( 1 + (-4.03 + 6.99i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.76 - 4.79i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 2.73T + 41T^{2} \)
43 \( 1 + 3.78T + 43T^{2} \)
47 \( 1 + (-5.53 - 9.58i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.93 - 10.2i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.734 + 1.27i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.53 + 9.58i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.18 - 10.7i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 7.06T + 71T^{2} \)
73 \( 1 + (2.76 - 4.79i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (8.32 + 14.4i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 1.46T + 83T^{2} \)
89 \( 1 + (-8.76 - 15.1i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 1.46T + 97T^{2} \)
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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

−9.213246452907090427335126501933, −8.684785554921938204642441760394, −7.83544731714904899342480310705, −6.45152721104872120039675557487, −6.26130119507787696896599313640, −5.81462253349931094178339368447, −4.20976590862400148885237439296, −3.28422314156270601589972307797, −2.66582786986717027584561526406, −1.31623566485156262946734278088, 0.815872663685776678219777722639, 1.73064732941473691623551030252, 3.13530182907505867319561331087, 4.26438405322413247403286438280, 4.77669610081969830296792030786, 5.86190296608137585459477088851, 6.67000706312051148447646747484, 7.23777947941940955510956107941, 8.524128511486702601054764984000, 8.967952820977056214009806410085

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