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Properties

Label 2-1008-63.25-c1-0-42
Degree $2$
Conductor $1008$
Sign $-0.998 + 0.0618i$
Analytic cond. $8.04892$
Root an. cond. $2.83706$
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  + (−0.748 − 1.56i)3-s + (2.11 − 3.65i)5-s + (−2.19 − 1.47i)7-s + (−1.88 + 2.33i)9-s + (0.964 + 1.67i)11-s + (−0.291 − 0.504i)13-s + (−7.29 − 0.562i)15-s + (3.61 − 6.25i)17-s + (−2.10 − 3.64i)19-s + (−0.670 + 4.53i)21-s + (0.639 − 1.10i)23-s + (−6.41 − 11.1i)25-s + (5.05 + 1.18i)27-s + (−4.20 + 7.27i)29-s + 0.952·31-s + ⋯
L(s)  = 1  + (−0.431 − 0.901i)3-s + (0.944 − 1.63i)5-s + (−0.829 − 0.559i)7-s + (−0.626 + 0.779i)9-s + (0.290 + 0.503i)11-s + (−0.0808 − 0.140i)13-s + (−1.88 − 0.145i)15-s + (0.875 − 1.51i)17-s + (−0.482 − 0.835i)19-s + (−0.146 + 0.989i)21-s + (0.133 − 0.231i)23-s + (−1.28 − 2.22i)25-s + (0.973 + 0.228i)27-s + (−0.780 + 1.35i)29-s + 0.171·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0618i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0618i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.998 + 0.0618i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ -0.998 + 0.0618i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.188703292\)
\(L(\frac12)\) \(\approx\) \(1.188703292\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.748 + 1.56i)T \)
7 \( 1 + (2.19 + 1.47i)T \)
good5 \( 1 + (-2.11 + 3.65i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.964 - 1.67i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.291 + 0.504i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.61 + 6.25i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.10 + 3.64i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.639 + 1.10i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.20 - 7.27i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 0.952T + 31T^{2} \)
37 \( 1 + (-3.03 - 5.25i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.31 - 2.27i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.442 - 0.766i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 5.76T + 47T^{2} \)
53 \( 1 + (0.962 - 1.66i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 4.55T + 59T^{2} \)
61 \( 1 + 10.5T + 61T^{2} \)
67 \( 1 - 4.86T + 67T^{2} \)
71 \( 1 + 11.5T + 71T^{2} \)
73 \( 1 + (-0.446 + 0.772i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 11.8T + 79T^{2} \)
83 \( 1 + (-5.24 + 9.08i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-3.87 - 6.71i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.98 - 3.44i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.462229623719192294732856135434, −8.884718520597774500883774131010, −7.79977420835945331161519295551, −6.94861259674094246986119783678, −6.15230713241870721249313073181, −5.19826334879756574941962088447, −4.62830545821609971708790416278, −2.88747900274214883299947964791, −1.55051834255551889151376865970, −0.57431097313190283311480768431, 2.14229508063059806595471304433, 3.29632040199527874010843669202, 3.85578149063030963860344863352, 5.64719305128344511676007311480, 6.02458436512339900602414144026, 6.56425042012604192421988474166, 7.86169571720474945599040993563, 9.071832057821327892219992151378, 9.795149627764571496236011946730, 10.27842439704994370212324836286

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