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Properties

Label 2-368-368.45-c2-0-68
Degree $2$
Conductor $368$
Sign $-0.0507 + 0.998i$
Analytic cond. $10.0272$
Root an. cond. $3.16658$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.24 + 1.56i)2-s + (−2.11 − 2.11i)3-s + (−0.879 + 3.90i)4-s + (0.661 − 5.94i)6-s + (−7.19 + 3.49i)8-s − 0.0498i·9-s + (10.1 − 6.39i)12-s + (−11.8 − 11.8i)13-s + (−14.4 − 6.86i)16-s + (0.0778 − 0.0622i)18-s − 23i·23-s + (22.6 + 7.81i)24-s − 25i·25-s + (3.71 − 33.3i)26-s + (−19.1 + 19.1i)27-s + ⋯
L(s)  = 1  + (0.624 + 0.780i)2-s + (−0.705 − 0.705i)3-s + (−0.219 + 0.975i)4-s + (0.110 − 0.991i)6-s + (−0.899 + 0.437i)8-s − 0.00553i·9-s + (0.842 − 0.532i)12-s + (−0.913 − 0.913i)13-s + (−0.903 − 0.429i)16-s + (0.00432 − 0.00345i)18-s − i·23-s + (0.942 + 0.325i)24-s − i·25-s + (0.142 − 1.28i)26-s + (−0.709 + 0.709i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0507 + 0.998i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0507 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(368\)    =    \(2^{4} \cdot 23\)
Sign: $-0.0507 + 0.998i$
Analytic conductor: \(10.0272\)
Root analytic conductor: \(3.16658\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{368} (45, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 368,\ (\ :1),\ -0.0507 + 0.998i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.554106 - 0.582961i\)
\(L(\frac12)\) \(\approx\) \(0.554106 - 0.582961i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.24 - 1.56i)T \)
23 \( 1 + 23iT \)
good3 \( 1 + (2.11 + 2.11i)T + 9iT^{2} \)
5 \( 1 + 25iT^{2} \)
7 \( 1 + 49T^{2} \)
11 \( 1 + 121iT^{2} \)
13 \( 1 + (11.8 + 11.8i)T + 169iT^{2} \)
17 \( 1 - 289T^{2} \)
19 \( 1 - 361iT^{2} \)
29 \( 1 + (21.7 + 21.7i)T + 841iT^{2} \)
31 \( 1 - 3.44T + 961T^{2} \)
37 \( 1 + 1.36e3iT^{2} \)
41 \( 1 + 66.2iT - 1.68e3T^{2} \)
43 \( 1 + 1.84e3iT^{2} \)
47 \( 1 - 50.9T + 2.20e3T^{2} \)
53 \( 1 + 2.80e3iT^{2} \)
59 \( 1 + (70.5 - 70.5i)T - 3.48e3iT^{2} \)
61 \( 1 - 3.72e3iT^{2} \)
67 \( 1 - 4.48e3iT^{2} \)
71 \( 1 - 130. iT - 5.04e3T^{2} \)
73 \( 1 - 88.0iT - 5.32e3T^{2} \)
79 \( 1 - 6.24e3T^{2} \)
83 \( 1 - 6.88e3iT^{2} \)
89 \( 1 + 7.92e3T^{2} \)
97 \( 1 - 9.40e3T^{2} \)
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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

−11.27431360763821531926509632342, −10.07781113930151333652661359875, −8.822839752146689953847050168986, −7.76869733305295835790393048309, −7.02071768501229947927334468747, −6.09829037558915523082281883075, −5.33270386748275085769821046417, −4.11957924471756594231697674071, −2.60091657204977130269684176864, −0.30827892041494805292051942711, 1.81044175088037394601691822361, 3.37429575763216231617853773000, 4.58027407698311188987284926201, 5.20412508063396707564765378031, 6.26480297311700598796453440967, 7.53860239510071134982584409205, 9.241813686532896030332239227226, 9.735802794842078149434821268344, 10.77967632268631822879150049915, 11.36485956401885727521002166028

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