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Properties

Label 2-704-88.5-c1-0-1
Degree $2$
Conductor $704$
Sign $-0.955 + 0.294i$
Analytic cond. $5.62146$
Root an. cond. $2.37096$
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  + (−1.21 + 1.67i)3-s + (−0.399 − 1.22i)9-s + (−2.90 + 1.59i)11-s + (0.472 − 1.45i)17-s + (−3.88 + 5.34i)19-s + (−4.04 − 2.93i)25-s + (−3.36 − 1.09i)27-s + (0.865 − 6.81i)33-s + (−7.20 − 5.23i)41-s − 4.97i·43-s + (−2.16 + 6.65i)49-s + (1.86 + 2.56i)51-s + (−4.22 − 13.0i)57-s + (−7.73 − 10.6i)59-s + 6.33i·67-s + ⋯
L(s)  = 1  + (−0.703 + 0.967i)3-s + (−0.133 − 0.409i)9-s + (−0.876 + 0.481i)11-s + (0.114 − 0.352i)17-s + (−0.890 + 1.22i)19-s + (−0.809 − 0.587i)25-s + (−0.647 − 0.210i)27-s + (0.150 − 1.18i)33-s + (−1.12 − 0.817i)41-s − 0.759i·43-s + (−0.309 + 0.951i)49-s + (0.260 + 0.358i)51-s + (−0.560 − 1.72i)57-s + (−1.00 − 1.38i)59-s + 0.774i·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.955 + 0.294i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.955 + 0.294i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(704\)    =    \(2^{6} \cdot 11\)
Sign: $-0.955 + 0.294i$
Analytic conductor: \(5.62146\)
Root analytic conductor: \(2.37096\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{704} (225, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 704,\ (\ :1/2),\ -0.955 + 0.294i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0546550 - 0.362700i\)
\(L(\frac12)\) \(\approx\) \(0.0546550 - 0.362700i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 + (2.90 - 1.59i)T \)
good3 \( 1 + (1.21 - 1.67i)T + (-0.927 - 2.85i)T^{2} \)
5 \( 1 + (4.04 + 2.93i)T^{2} \)
7 \( 1 + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (-0.472 + 1.45i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (3.88 - 5.34i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + (-8.96 + 27.5i)T^{2} \)
31 \( 1 + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (7.20 + 5.23i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 4.97iT - 43T^{2} \)
47 \( 1 + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (7.73 + 10.6i)T + (-18.2 + 56.1i)T^{2} \)
61 \( 1 + (49.3 + 35.8i)T^{2} \)
67 \( 1 - 6.33iT - 67T^{2} \)
71 \( 1 + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (13.5 - 9.85i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-2.73 - 0.887i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 - 10.9T + 89T^{2} \)
97 \( 1 + (-5.94 - 18.2i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.59467047579657525054906357999, −10.28578154358023310731678389339, −9.480516582745121084829472747371, −8.320247568570431877466213769763, −7.50225207608375079418102350629, −6.25136081770007379798025949338, −5.41165630573685596505558074467, −4.60555998704586851554260084710, −3.67842080545398110058933094412, −2.13070108179221010863417472521, 0.19906754559800217465764310062, 1.75678854797318774708224405510, 3.12481766522798526255852653144, 4.61050306833246615084562412182, 5.67331710627120989927898461039, 6.39473024808078554764867562821, 7.26357856326842620316659148137, 8.070268514695046151215770068637, 9.013381910226324277696699790331, 10.13370327296339987378750328130

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