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Properties

Label 2-2016-56.37-c1-0-37
Degree $2$
Conductor $2016$
Sign $-0.550 - 0.834i$
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  + (−3.09 − 1.78i)5-s + (−0.993 − 2.45i)7-s + (−0.815 + 0.470i)11-s − 6.15i·13-s + (−1.89 − 3.27i)17-s + (−2.09 − 1.20i)19-s + (−1.49 + 2.58i)23-s + (3.90 + 6.75i)25-s − 2.68i·29-s + (5.35 + 9.27i)31-s + (−1.30 + 9.37i)35-s + (−1.47 − 0.853i)37-s + 4.55·41-s + 3.50i·43-s + (3.42 − 5.92i)47-s + ⋯
L(s)  = 1  + (−1.38 − 0.800i)5-s + (−0.375 − 0.926i)7-s + (−0.245 + 0.142i)11-s − 1.70i·13-s + (−0.458 − 0.794i)17-s + (−0.479 − 0.276i)19-s + (−0.311 + 0.539i)23-s + (0.780 + 1.35i)25-s − 0.497i·29-s + (0.962 + 1.66i)31-s + (−0.221 + 1.58i)35-s + (−0.243 − 0.140i)37-s + 0.712·41-s + 0.534i·43-s + (0.499 − 0.864i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.550 - 0.834i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.550 - 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2016\)    =    \(2^{5} \cdot 3^{2} \cdot 7\)
Sign: $-0.550 - 0.834i$
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: \((2,\ 2016,\ (\ :1/2),\ -0.550 - 0.834i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2189902541\)
\(L(\frac12)\) \(\approx\) \(0.2189902541\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.993 + 2.45i)T \)
good5 \( 1 + (3.09 + 1.78i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.815 - 0.470i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 6.15iT - 13T^{2} \)
17 \( 1 + (1.89 + 3.27i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.09 + 1.20i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.49 - 2.58i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 2.68iT - 29T^{2} \)
31 \( 1 + (-5.35 - 9.27i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.47 + 0.853i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 4.55T + 41T^{2} \)
43 \( 1 - 3.50iT - 43T^{2} \)
47 \( 1 + (-3.42 + 5.92i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.57 - 3.79i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.100 + 0.0580i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.06 + 4.07i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.44 + 1.98i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 3.92T + 71T^{2} \)
73 \( 1 + (3.11 + 5.39i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.73 - 4.73i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 1.19iT - 83T^{2} \)
89 \( 1 + (0.910 - 1.57i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 12.0T + 97T^{2} \)
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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

−8.469360453646926121215598893786, −7.80363454787156933830160529772, −7.36439248776337786132452335751, −6.36228938578472802156740149830, −5.14787700090367292553681660381, −4.55673175891771799386234746683, −3.66628838584589434476343249611, −2.88124830988028118405531493210, −0.987345270150443266771459939789, −0.095540317539969997267137557944, 2.04145193610057444723399261768, 2.94919844914497573069756586049, 4.02037674503992543534092747350, 4.48229280632995034050460360217, 5.97788128749959610023222458700, 6.50579238197460407885962899498, 7.30450565903187825700262694589, 8.167131384333525186774040394773, 8.745335682643055097290101004844, 9.569082324457402548038006895224

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