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Properties

Label 2-182-91.80-c1-0-1
Degree $2$
Conductor $182$
Sign $0.751 - 0.659i$
Analytic cond. $1.45327$
Root an. cond. $1.20551$
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  + (−0.258 − 0.965i)2-s + 0.0362i·3-s + (−0.866 + 0.499i)4-s + (−0.869 + 3.24i)5-s + (0.0350 − 0.00938i)6-s + (−0.965 + 2.46i)7-s + (0.707 + 0.707i)8-s + 2.99·9-s + 3.35·10-s + (2.20 + 2.20i)11-s + (−0.0181 − 0.0314i)12-s + (−3.42 − 1.12i)13-s + (2.62 + 0.295i)14-s + (−0.117 − 0.0315i)15-s + (0.500 − 0.866i)16-s + (−2.97 − 5.14i)17-s + ⋯
L(s)  = 1  + (−0.183 − 0.683i)2-s + 0.0209i·3-s + (−0.433 + 0.249i)4-s + (−0.388 + 1.45i)5-s + (0.0142 − 0.00383i)6-s + (−0.365 + 0.931i)7-s + (0.249 + 0.249i)8-s + 0.999·9-s + 1.06·10-s + (0.664 + 0.664i)11-s + (−0.00523 − 0.00906i)12-s + (−0.950 − 0.310i)13-s + (0.702 + 0.0789i)14-s + (−0.0303 − 0.00814i)15-s + (0.125 − 0.216i)16-s + (−0.721 − 1.24i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 182 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.751 - 0.659i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 182 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.751 - 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(182\)    =    \(2 \cdot 7 \cdot 13\)
Sign: $0.751 - 0.659i$
Analytic conductor: \(1.45327\)
Root analytic conductor: \(1.20551\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{182} (171, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 182,\ (\ :1/2),\ 0.751 - 0.659i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.874200 + 0.329254i\)
\(L(\frac12)\) \(\approx\) \(0.874200 + 0.329254i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 + 0.965i)T \)
7 \( 1 + (0.965 - 2.46i)T \)
13 \( 1 + (3.42 + 1.12i)T \)
good3 \( 1 - 0.0362iT - 3T^{2} \)
5 \( 1 + (0.869 - 3.24i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-2.20 - 2.20i)T + 11iT^{2} \)
17 \( 1 + (2.97 + 5.14i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.46 - 1.46i)T + 19iT^{2} \)
23 \( 1 + (-5.64 - 3.25i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.68 - 4.64i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.23 - 0.598i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (-2.90 + 0.779i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-1.87 + 7.00i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (6.96 + 4.01i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.13 - 1.37i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-2.87 + 4.98i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-11.0 - 2.95i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + 3.25iT - 61T^{2} \)
67 \( 1 + (4.83 - 4.83i)T - 67iT^{2} \)
71 \( 1 + (4.14 + 15.4i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (3.01 + 11.2i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-4.09 - 7.09i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-9.96 - 9.96i)T + 83iT^{2} \)
89 \( 1 + (-2.92 - 10.9i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-1.69 + 0.454i)T + (84.0 - 48.5i)T^{2} \)
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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

−12.42514493302479593442208895636, −11.79761080758873687285229125467, −10.78564207804473465303843019423, −9.840303797038106690610880915343, −9.114769357082250697096464705527, −7.35296085560538202361029581039, −6.86273354947682703484311628044, −5.00955824883839321965542000230, −3.47764137587245058482469586894, −2.35149291474644100046032271846, 0.996791552275835481040582291228, 4.09935048891590543408422340515, 4.74977015194718245138192341441, 6.39057938118496265423613575836, 7.37520015381798572864476865192, 8.461777723122421312127672028945, 9.316197888695513443084700313222, 10.29322256658534300512540484434, 11.66228935095520309402353818511, 12.97586867613328827390439877789

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