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Properties

Label 2-187-17.4-c1-0-15
Degree $2$
Conductor $187$
Sign $-0.996 - 0.0781i$
Analytic cond. $1.49320$
Root an. cond. $1.22196$
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  − 2.71i·2-s + (2.10 − 2.10i)3-s − 5.39·4-s + (−0.643 + 0.643i)5-s + (−5.71 − 5.71i)6-s + (1.95 + 1.95i)7-s + 9.21i·8-s − 5.83i·9-s + (1.75 + 1.75i)10-s + (−0.707 − 0.707i)11-s + (−11.3 + 11.3i)12-s − 0.843·13-s + (5.30 − 5.30i)14-s + 2.70i·15-s + 14.2·16-s + (4.11 − 0.182i)17-s + ⋯
L(s)  = 1  − 1.92i·2-s + (1.21 − 1.21i)3-s − 2.69·4-s + (−0.287 + 0.287i)5-s + (−2.33 − 2.33i)6-s + (0.737 + 0.737i)7-s + 3.25i·8-s − 1.94i·9-s + (0.553 + 0.553i)10-s + (−0.213 − 0.213i)11-s + (−3.26 + 3.26i)12-s − 0.233·13-s + (1.41 − 1.41i)14-s + 0.698i·15-s + 3.56·16-s + (0.999 − 0.0442i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0781i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 - 0.0781i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(187\)    =    \(11 \cdot 17\)
Sign: $-0.996 - 0.0781i$
Analytic conductor: \(1.49320\)
Root analytic conductor: \(1.22196\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{187} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 187,\ (\ :1/2),\ -0.996 - 0.0781i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0555215 + 1.41948i\)
\(L(\frac12)\) \(\approx\) \(0.0555215 + 1.41948i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (0.707 + 0.707i)T \)
17 \( 1 + (-4.11 + 0.182i)T \)
good2 \( 1 + 2.71iT - 2T^{2} \)
3 \( 1 + (-2.10 + 2.10i)T - 3iT^{2} \)
5 \( 1 + (0.643 - 0.643i)T - 5iT^{2} \)
7 \( 1 + (-1.95 - 1.95i)T + 7iT^{2} \)
13 \( 1 + 0.843T + 13T^{2} \)
19 \( 1 + 2.73iT - 19T^{2} \)
23 \( 1 + (-0.144 - 0.144i)T + 23iT^{2} \)
29 \( 1 + (-3.87 + 3.87i)T - 29iT^{2} \)
31 \( 1 + (7.72 - 7.72i)T - 31iT^{2} \)
37 \( 1 + (7.21 - 7.21i)T - 37iT^{2} \)
41 \( 1 + (-0.838 - 0.838i)T + 41iT^{2} \)
43 \( 1 + 5.42iT - 43T^{2} \)
47 \( 1 - 6.16T + 47T^{2} \)
53 \( 1 - 2.19iT - 53T^{2} \)
59 \( 1 + 4.42iT - 59T^{2} \)
61 \( 1 + (-5.39 - 5.39i)T + 61iT^{2} \)
67 \( 1 + 2.90T + 67T^{2} \)
71 \( 1 + (-0.772 + 0.772i)T - 71iT^{2} \)
73 \( 1 + (4.29 - 4.29i)T - 73iT^{2} \)
79 \( 1 + (4.55 + 4.55i)T + 79iT^{2} \)
83 \( 1 - 1.77iT - 83T^{2} \)
89 \( 1 + 11.4T + 89T^{2} \)
97 \( 1 + (-6.06 + 6.06i)T - 97iT^{2} \)
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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

−12.12755045697110032006288116915, −11.43177594262103758652616742460, −10.22010340131295524861818088443, −8.973430773243285306909973423432, −8.465896863368233279466492359277, −7.38837958336783990355347933673, −5.22686585167285741288064322979, −3.48734849014432979945109380036, −2.60667338208357767590962949737, −1.47552519277420865883756510438, 3.77335843330328101400330014447, 4.52281388546786487142083539017, 5.52178350187449355528786085547, 7.36612163181013787893916642928, 7.963278139603275426938892833230, 8.744077219797277431942607232240, 9.683134677825422945732375805740, 10.52614980906998418299102792731, 12.62093984998593851165544116808, 13.84604719429101790450644861238

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