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Properties

Label 2-187-17.4-c1-0-1
Degree $2$
Conductor $187$
Sign $-0.484 + 0.875i$
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  + 2.37i·2-s + (−1.69 + 1.69i)3-s − 3.62·4-s + (−0.324 + 0.324i)5-s + (−4.01 − 4.01i)6-s + (1.08 + 1.08i)7-s − 3.84i·8-s − 2.73i·9-s + (−0.768 − 0.768i)10-s + (−0.707 − 0.707i)11-s + (6.12 − 6.12i)12-s + 1.06·13-s + (−2.57 + 2.57i)14-s − 1.09i·15-s + 1.86·16-s + (1.53 − 3.82i)17-s + ⋯
L(s)  = 1  + 1.67i·2-s + (−0.977 + 0.977i)3-s − 1.81·4-s + (−0.144 + 0.144i)5-s + (−1.63 − 1.63i)6-s + (0.410 + 0.410i)7-s − 1.35i·8-s − 0.910i·9-s + (−0.242 − 0.242i)10-s + (−0.213 − 0.213i)11-s + (1.76 − 1.76i)12-s + 0.295·13-s + (−0.688 + 0.688i)14-s − 0.283i·15-s + 0.466·16-s + (0.373 − 0.927i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.484 + 0.875i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.484 + 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(187\)    =    \(11 \cdot 17\)
Sign: $-0.484 + 0.875i$
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: \((2,\ 187,\ (\ :1/2),\ -0.484 + 0.875i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.337975 - 0.573164i\)
\(L(\frac12)\) \(\approx\) \(0.337975 - 0.573164i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.53 + 3.82i)T \)
good2 \( 1 - 2.37iT - 2T^{2} \)
3 \( 1 + (1.69 - 1.69i)T - 3iT^{2} \)
5 \( 1 + (0.324 - 0.324i)T - 5iT^{2} \)
7 \( 1 + (-1.08 - 1.08i)T + 7iT^{2} \)
13 \( 1 - 1.06T + 13T^{2} \)
19 \( 1 - 6.52iT - 19T^{2} \)
23 \( 1 + (0.493 + 0.493i)T + 23iT^{2} \)
29 \( 1 + (-0.169 + 0.169i)T - 29iT^{2} \)
31 \( 1 + (5.69 - 5.69i)T - 31iT^{2} \)
37 \( 1 + (1.66 - 1.66i)T - 37iT^{2} \)
41 \( 1 + (2.72 + 2.72i)T + 41iT^{2} \)
43 \( 1 - 3.22iT - 43T^{2} \)
47 \( 1 - 10.3T + 47T^{2} \)
53 \( 1 + 1.24iT - 53T^{2} \)
59 \( 1 - 10.1iT - 59T^{2} \)
61 \( 1 + (-9.61 - 9.61i)T + 61iT^{2} \)
67 \( 1 - 14.8T + 67T^{2} \)
71 \( 1 + (-2.38 + 2.38i)T - 71iT^{2} \)
73 \( 1 + (0.254 - 0.254i)T - 73iT^{2} \)
79 \( 1 + (0.436 + 0.436i)T + 79iT^{2} \)
83 \( 1 - 11.0iT - 83T^{2} \)
89 \( 1 - 13.7T + 89T^{2} \)
97 \( 1 + (-4.11 + 4.11i)T - 97iT^{2} \)
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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

−13.64599841095950856939279331788, −12.18972365132806073604874773793, −11.19857264583823755517255937847, −10.15421284235317201246722315459, −9.061775574896507224764046582495, −8.023736086039533940890803510329, −6.92572249150939594916410507301, −5.57831096576664333771928322045, −5.31413286764615026659722829548, −3.93239568467142251141876138689, 0.69833244350599368282186931237, 2.09841243625747660695872852319, 3.92315245571527064146084746850, 5.20267782240866814897089861941, 6.64286027197311110002446835472, 7.914914903956869284930053655145, 9.222888043957124286595645120888, 10.49706453195737788804985988948, 11.13569040222888435174362220396, 11.84529688493942471899955573661

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