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Properties

Label 2-182-13.9-c3-0-12
Degree $2$
Conductor $182$
Sign $0.999 + 0.0356i$
Analytic cond. $10.7383$
Root an. cond. $3.27694$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + 1.73i)2-s + (2.60 − 4.50i)3-s + (−1.99 − 3.46i)4-s + 17.5·5-s + (5.20 + 9.01i)6-s + (3.5 + 6.06i)7-s + 7.99·8-s + (−0.0347 − 0.0601i)9-s + (−17.5 + 30.4i)10-s + (17.7 − 30.7i)11-s − 20.8·12-s + (−13.4 + 44.9i)13-s − 14·14-s + (45.6 − 79.1i)15-s + (−8 + 13.8i)16-s + (9.71 + 16.8i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.500 − 0.867i)3-s + (−0.249 − 0.433i)4-s + 1.57·5-s + (0.354 + 0.613i)6-s + (0.188 + 0.327i)7-s + 0.353·8-s + (−0.00128 − 0.00222i)9-s + (−0.555 + 0.961i)10-s + (0.487 − 0.843i)11-s − 0.500·12-s + (−0.286 + 0.957i)13-s − 0.267·14-s + (0.786 − 1.36i)15-s + (−0.125 + 0.216i)16-s + (0.138 + 0.240i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 182 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0356i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 182 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0356i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(182\)    =    \(2 \cdot 7 \cdot 13\)
Sign: $0.999 + 0.0356i$
Analytic conductor: \(10.7383\)
Root analytic conductor: \(3.27694\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{182} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 182,\ (\ :3/2),\ 0.999 + 0.0356i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.24722 - 0.0400783i\)
\(L(\frac12)\) \(\approx\) \(2.24722 - 0.0400783i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 - 1.73i)T \)
7 \( 1 + (-3.5 - 6.06i)T \)
13 \( 1 + (13.4 - 44.9i)T \)
good3 \( 1 + (-2.60 + 4.50i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 - 17.5T + 125T^{2} \)
11 \( 1 + (-17.7 + 30.7i)T + (-665.5 - 1.15e3i)T^{2} \)
17 \( 1 + (-9.71 - 16.8i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (36.3 + 62.9i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-27.6 + 47.9i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-62.5 + 108. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 - 71.0T + 2.97e4T^{2} \)
37 \( 1 + (176. - 305. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-120. + 208. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-122. - 212. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + 237.T + 1.03e5T^{2} \)
53 \( 1 + 152.T + 1.48e5T^{2} \)
59 \( 1 + (12.2 + 21.3i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (357. + 618. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-480. + 831. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (-229. - 396. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + 100.T + 3.89e5T^{2} \)
79 \( 1 + 1.14e3T + 4.93e5T^{2} \)
83 \( 1 + 861.T + 5.71e5T^{2} \)
89 \( 1 + (528. - 915. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-451. - 782. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−12.47180409353202582893440599951, −11.05948500614098115469376263956, −9.854359359701930111443333748544, −8.986368758812658658443558153051, −8.196249176139804585646991556316, −6.76355612285631821967500892289, −6.22454349564373634816202637856, −4.88422228892315953685597458861, −2.48068893447024774087466914721, −1.38818199241727884810794457344, 1.51010222727586527376475283357, 2.91027735556029368286404825248, 4.30800809896049979471500002116, 5.57301650883774681856090381082, 7.07733571693540009494389422883, 8.581777498358123929173106233452, 9.545655074341899488754660184152, 10.02469883358690293372986251875, 10.71131622628808840273814748190, 12.28164517002166272104790731853

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