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Properties

Label 2-182-91.23-c1-0-9
Degree $2$
Conductor $182$
Sign $-0.710 + 0.703i$
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  − i·2-s + (1.24 − 2.16i)3-s − 4-s + (−2.41 − 1.39i)5-s + (−2.16 − 1.24i)6-s + (2.31 + 1.27i)7-s + i·8-s + (−1.61 − 2.79i)9-s + (−1.39 + 2.41i)10-s + (−3.26 − 1.88i)11-s + (−1.24 + 2.16i)12-s + (3.40 + 1.19i)13-s + (1.27 − 2.31i)14-s + (−6.02 + 3.47i)15-s + 16-s + 5.01·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.720 − 1.24i)3-s − 0.5·4-s + (−1.07 − 0.623i)5-s + (−0.882 − 0.509i)6-s + (0.876 + 0.481i)7-s + 0.353i·8-s + (−0.538 − 0.932i)9-s + (−0.440 + 0.763i)10-s + (−0.984 − 0.568i)11-s + (−0.360 + 0.623i)12-s + (0.943 + 0.331i)13-s + (0.340 − 0.619i)14-s + (−1.55 + 0.897i)15-s + 0.250·16-s + 1.21·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 182 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.710 + 0.703i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 182 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.710 + 0.703i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(182\)    =    \(2 \cdot 7 \cdot 13\)
Sign: $-0.710 + 0.703i$
Analytic conductor: \(1.45327\)
Root analytic conductor: \(1.20551\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{182} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 182,\ (\ :1/2),\ -0.710 + 0.703i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.469170 - 1.14026i\)
\(L(\frac12)\) \(\approx\) \(0.469170 - 1.14026i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
7 \( 1 + (-2.31 - 1.27i)T \)
13 \( 1 + (-3.40 - 1.19i)T \)
good3 \( 1 + (-1.24 + 2.16i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (2.41 + 1.39i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (3.26 + 1.88i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 - 5.01T + 17T^{2} \)
19 \( 1 + (2.68 - 1.54i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 0.0974T + 23T^{2} \)
29 \( 1 + (1.34 + 2.33i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-7.13 + 4.12i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 0.489iT - 37T^{2} \)
41 \( 1 + (8.29 - 4.79i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.642 + 1.11i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-8.54 - 4.93i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.13 - 8.90i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 5.43iT - 59T^{2} \)
61 \( 1 + (-1.31 - 2.27i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.32 + 4.22i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (5.07 + 2.92i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (13.2 - 7.66i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.55 - 9.62i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 16.0iT - 83T^{2} \)
89 \( 1 - 14.3iT - 89T^{2} \)
97 \( 1 + (-5.50 - 3.18i)T + (48.5 + 84.0i)T^{2} \)
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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

−12.14006975817690956878442473272, −11.72424294972458827442500751833, −10.52084610616163744643226200494, −8.787885161495363739352753417578, −8.181508330287261663347153287201, −7.67908220663672363781677742974, −5.81099140343772017498535965590, −4.27700816809727866520378779577, −2.78597619416249499745863886096, −1.24355460070666190980596870191, 3.27132081817484542358132575253, 4.21285159166468951239787955156, 5.24325700693590282637360225686, 7.12095210506386887754649931392, 8.033272964228212492816331485771, 8.665887704647395545383276943805, 10.23324250082603310707750662931, 10.61159114742037958493020138742, 11.88501701499072334418345224854, 13.36965572770189425672840944656

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