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Properties

Label 2-182-7.2-c1-0-3
Degree $2$
Conductor $182$
Sign $0.860 - 0.509i$
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  + (−0.5 + 0.866i)2-s + (0.529 + 0.916i)3-s + (−0.499 − 0.866i)4-s + (1.96 − 3.41i)5-s − 1.05·6-s + (1.20 + 2.35i)7-s + 0.999·8-s + (0.940 − 1.62i)9-s + (1.96 + 3.41i)10-s + (−0.326 − 0.566i)11-s + (0.529 − 0.916i)12-s − 13-s + (−2.64 − 0.137i)14-s + 4.16·15-s + (−0.5 + 0.866i)16-s + (1.11 + 1.92i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.305 + 0.529i)3-s + (−0.249 − 0.433i)4-s + (0.880 − 1.52i)5-s − 0.432·6-s + (0.454 + 0.890i)7-s + 0.353·8-s + (0.313 − 0.542i)9-s + (0.622 + 1.07i)10-s + (−0.0985 − 0.170i)11-s + (0.152 − 0.264i)12-s − 0.277·13-s + (−0.706 − 0.0367i)14-s + 1.07·15-s + (−0.125 + 0.216i)16-s + (0.269 + 0.467i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 182 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.860 - 0.509i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 182 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.860 - 0.509i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(182\)    =    \(2 \cdot 7 \cdot 13\)
Sign: $0.860 - 0.509i$
Analytic conductor: \(1.45327\)
Root analytic conductor: \(1.20551\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{182} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 182,\ (\ :1/2),\ 0.860 - 0.509i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.18566 + 0.324980i\)
\(L(\frac12)\) \(\approx\) \(1.18566 + 0.324980i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (-1.20 - 2.35i)T \)
13 \( 1 + T \)
good3 \( 1 + (-0.529 - 0.916i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.96 + 3.41i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.326 + 0.566i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.11 - 1.92i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.55 - 6.15i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.05 - 3.56i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 4.65T + 29T^{2} \)
31 \( 1 + (5.11 + 8.85i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.529 + 0.916i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 7.64T + 41T^{2} \)
43 \( 1 + 4.05T + 43T^{2} \)
47 \( 1 + (-2.11 + 3.65i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.61 - 7.98i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.38 + 5.86i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.66 - 8.08i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.326 + 0.566i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 5.53T + 71T^{2} \)
73 \( 1 + (3.46 + 5.99i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.46 - 5.99i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (2.46 - 4.26i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 9.87T + 97T^{2} \)
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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

−12.71018568220194025416924858869, −12.00619997079303548586606757347, −10.23508420169170353909895654995, −9.503452431651975444246673701189, −8.741890107089441715330259491811, −8.030563458186866504183135175743, −6.10657709977702482876091843701, −5.38273026569341791135827096479, −4.18946248223678205048472693830, −1.72006658413250223722067065420, 1.92632530660567977504350491177, 3.00002728251666230586463215535, 4.79196349400591361513357494730, 6.76782842736139804214741699116, 7.22617084883821239220199383815, 8.494938735274601488651868445530, 9.976334327675847503421398306031, 10.52663979905351515561724992621, 11.23105973877818091655549388963, 12.64767654107879048298079350122

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