Properties

Label 2-736-184.133-c1-0-2
Degree $2$
Conductor $736$
Sign $0.315 - 0.948i$
Analytic cond. $5.87698$
Root an. cond. $2.42425$
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  + (0.457 + 0.396i)3-s + (−3.79 − 0.545i)5-s + (−0.480 − 1.05i)7-s + (−0.374 − 2.60i)9-s + (1.35 + 4.62i)11-s + (5.63 + 2.57i)13-s + (−1.51 − 1.75i)15-s + (−0.761 + 0.489i)17-s + (−2.21 + 3.44i)19-s + (0.197 − 0.672i)21-s + (−0.624 + 4.75i)23-s + (9.29 + 2.72i)25-s + (1.84 − 2.86i)27-s + (2.70 + 4.20i)29-s + (4.49 + 5.19i)31-s + ⋯
L(s)  = 1  + (0.264 + 0.228i)3-s + (−1.69 − 0.243i)5-s + (−0.181 − 0.398i)7-s + (−0.124 − 0.868i)9-s + (0.409 + 1.39i)11-s + (1.56 + 0.714i)13-s + (−0.392 − 0.452i)15-s + (−0.184 + 0.118i)17-s + (−0.508 + 0.791i)19-s + (0.0430 − 0.146i)21-s + (−0.130 + 0.991i)23-s + (1.85 + 0.545i)25-s + (0.354 − 0.552i)27-s + (0.501 + 0.781i)29-s + (0.807 + 0.932i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.315 - 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.315 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(736\)    =    \(2^{5} \cdot 23\)
Sign: $0.315 - 0.948i$
Analytic conductor: \(5.87698\)
Root analytic conductor: \(2.42425\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{736} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 736,\ (\ :1/2),\ 0.315 - 0.948i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.849603 + 0.612919i\)
\(L(\frac12)\) \(\approx\) \(0.849603 + 0.612919i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
23 \( 1 + (0.624 - 4.75i)T \)
good3 \( 1 + (-0.457 - 0.396i)T + (0.426 + 2.96i)T^{2} \)
5 \( 1 + (3.79 + 0.545i)T + (4.79 + 1.40i)T^{2} \)
7 \( 1 + (0.480 + 1.05i)T + (-4.58 + 5.29i)T^{2} \)
11 \( 1 + (-1.35 - 4.62i)T + (-9.25 + 5.94i)T^{2} \)
13 \( 1 + (-5.63 - 2.57i)T + (8.51 + 9.82i)T^{2} \)
17 \( 1 + (0.761 - 0.489i)T + (7.06 - 15.4i)T^{2} \)
19 \( 1 + (2.21 - 3.44i)T + (-7.89 - 17.2i)T^{2} \)
29 \( 1 + (-2.70 - 4.20i)T + (-12.0 + 26.3i)T^{2} \)
31 \( 1 + (-4.49 - 5.19i)T + (-4.41 + 30.6i)T^{2} \)
37 \( 1 + (5.93 - 0.852i)T + (35.5 - 10.4i)T^{2} \)
41 \( 1 + (0.166 - 1.15i)T + (-39.3 - 11.5i)T^{2} \)
43 \( 1 + (-2.41 - 2.09i)T + (6.11 + 42.5i)T^{2} \)
47 \( 1 + 5.80T + 47T^{2} \)
53 \( 1 + (-0.959 + 0.438i)T + (34.7 - 40.0i)T^{2} \)
59 \( 1 + (10.8 + 4.94i)T + (38.6 + 44.5i)T^{2} \)
61 \( 1 + (-2.03 + 1.76i)T + (8.68 - 60.3i)T^{2} \)
67 \( 1 + (0.716 - 2.44i)T + (-56.3 - 36.2i)T^{2} \)
71 \( 1 + (-12.4 - 3.64i)T + (59.7 + 38.3i)T^{2} \)
73 \( 1 + (-6.67 - 4.28i)T + (30.3 + 66.4i)T^{2} \)
79 \( 1 + (-1.71 + 3.74i)T + (-51.7 - 59.7i)T^{2} \)
83 \( 1 + (-2.18 + 0.313i)T + (79.6 - 23.3i)T^{2} \)
89 \( 1 + (2.22 - 2.56i)T + (-12.6 - 88.0i)T^{2} \)
97 \( 1 + (0.561 - 3.90i)T + (-93.0 - 27.3i)T^{2} \)
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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

−10.61407139054352034985208342422, −9.591701151270298330117911160195, −8.715868211742303840078117387108, −8.120573591291224035073977932264, −7.02604461083925127299932559967, −6.45921304504051323579722975414, −4.75959796420059630357203335412, −3.86986028051491852908979073166, −3.51924026194414728121590365749, −1.34850791481724921693482811237, 0.59649448344362457845237752462, 2.71841905423327319911327810094, 3.57218367845025388222379956586, 4.52279201046970687350557283155, 5.89806246170240617830408246173, 6.73317626881928668799763512103, 7.965402812842671510161563697842, 8.312550889017749861060803203158, 8.930000146933438962661571603393, 10.67585062687505485309648780219

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