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Properties

Label 2-1008-63.58-c1-0-35
Degree $2$
Conductor $1008$
Sign $0.281 + 0.959i$
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  + (1.13 − 1.30i)3-s + (−0.170 − 0.294i)5-s + (2.63 + 0.253i)7-s + (−0.429 − 2.96i)9-s + (−0.335 + 0.581i)11-s + (1.62 − 2.81i)13-s + (−0.578 − 0.111i)15-s + (−1.10 − 1.90i)17-s + (−0.242 + 0.419i)19-s + (3.31 − 3.16i)21-s + (2.09 + 3.62i)23-s + (2.44 − 4.22i)25-s + (−4.37 − 2.80i)27-s + (0.478 + 0.829i)29-s − 2.08·31-s + ⋯
L(s)  = 1  + (0.654 − 0.756i)3-s + (−0.0760 − 0.131i)5-s + (0.995 + 0.0957i)7-s + (−0.143 − 0.989i)9-s + (−0.101 + 0.175i)11-s + (0.450 − 0.779i)13-s + (−0.149 − 0.0287i)15-s + (−0.266 − 0.462i)17-s + (−0.0555 + 0.0961i)19-s + (0.723 − 0.689i)21-s + (0.436 + 0.756i)23-s + (0.488 − 0.845i)25-s + (−0.842 − 0.539i)27-s + (0.0889 + 0.154i)29-s − 0.374·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.281 + 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.281 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.281 + 0.959i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (625, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.281 + 0.959i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.164178071\)
\(L(\frac12)\) \(\approx\) \(2.164178071\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.13 + 1.30i)T \)
7 \( 1 + (-2.63 - 0.253i)T \)
good5 \( 1 + (0.170 + 0.294i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.335 - 0.581i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.62 + 2.81i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.10 + 1.90i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.242 - 0.419i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.09 - 3.62i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.478 - 0.829i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 2.08T + 31T^{2} \)
37 \( 1 + (-4.81 + 8.34i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.90 - 6.75i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.66 - 6.34i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 2.69T + 47T^{2} \)
53 \( 1 + (6.12 + 10.6i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 4.94T + 59T^{2} \)
61 \( 1 + 3.52T + 61T^{2} \)
67 \( 1 + 12.3T + 67T^{2} \)
71 \( 1 - 5.57T + 71T^{2} \)
73 \( 1 + (3.71 + 6.43i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 10.0T + 79T^{2} \)
83 \( 1 + (2.47 + 4.28i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (8.52 - 14.7i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.23 - 7.33i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.579004041397808284236423864463, −8.815571086678606294058434116434, −8.026807169452000742127607575267, −7.54628813989480490067510469659, −6.51122471697879618223445607010, −5.51110436860261406764566782210, −4.48618820042429299316932856989, −3.27700859574577318303140890817, −2.20763161027887891071316387090, −1.01397173827158113032369188246, 1.65490456498437462151412113111, 2.84938339498918275525262242700, 4.02052174571768677002503150956, 4.67580249698600053491218061691, 5.66594946526298509028811722994, 6.91036193893597540510028658893, 7.80650059674256050026080777317, 8.670441668808630011938013433105, 9.066130176902048559051507214461, 10.22160695231456818725858412169

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