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Properties

Label 2-187-17.13-c1-0-8
Degree $2$
Conductor $187$
Sign $0.993 + 0.114i$
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  − 1.19i·2-s + (1.79 + 1.79i)3-s + 0.576·4-s + (0.279 + 0.279i)5-s + (2.13 − 2.13i)6-s + (−2.05 + 2.05i)7-s − 3.07i·8-s + 3.42i·9-s + (0.333 − 0.333i)10-s + (0.707 − 0.707i)11-s + (1.03 + 1.03i)12-s − 5.12·13-s + (2.44 + 2.44i)14-s + 1.00i·15-s − 2.51·16-s + (4.00 + 0.968i)17-s + ⋯
L(s)  = 1  − 0.843i·2-s + (1.03 + 1.03i)3-s + 0.288·4-s + (0.125 + 0.125i)5-s + (0.872 − 0.872i)6-s + (−0.775 + 0.775i)7-s − 1.08i·8-s + 1.14i·9-s + (0.105 − 0.105i)10-s + (0.213 − 0.213i)11-s + (0.298 + 0.298i)12-s − 1.42·13-s + (0.654 + 0.654i)14-s + 0.258i·15-s − 0.628·16-s + (0.972 + 0.234i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.114i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 + 0.114i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(187\)    =    \(11 \cdot 17\)
Sign: $0.993 + 0.114i$
Analytic conductor: \(1.49320\)
Root analytic conductor: \(1.22196\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{187} (166, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 187,\ (\ :1/2),\ 0.993 + 0.114i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.64962 - 0.0946550i\)
\(L(\frac12)\) \(\approx\) \(1.64962 - 0.0946550i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.00 - 0.968i)T \)
good2 \( 1 + 1.19iT - 2T^{2} \)
3 \( 1 + (-1.79 - 1.79i)T + 3iT^{2} \)
5 \( 1 + (-0.279 - 0.279i)T + 5iT^{2} \)
7 \( 1 + (2.05 - 2.05i)T - 7iT^{2} \)
13 \( 1 + 5.12T + 13T^{2} \)
19 \( 1 + 6.66iT - 19T^{2} \)
23 \( 1 + (1.60 - 1.60i)T - 23iT^{2} \)
29 \( 1 + (-0.891 - 0.891i)T + 29iT^{2} \)
31 \( 1 + (5.00 + 5.00i)T + 31iT^{2} \)
37 \( 1 + (-5.43 - 5.43i)T + 37iT^{2} \)
41 \( 1 + (5.08 - 5.08i)T - 41iT^{2} \)
43 \( 1 - 7.26iT - 43T^{2} \)
47 \( 1 - 1.47T + 47T^{2} \)
53 \( 1 - 8.48iT - 53T^{2} \)
59 \( 1 - 0.787iT - 59T^{2} \)
61 \( 1 + (4.23 - 4.23i)T - 61iT^{2} \)
67 \( 1 + 11.0T + 67T^{2} \)
71 \( 1 + (-2.11 - 2.11i)T + 71iT^{2} \)
73 \( 1 + (-3.26 - 3.26i)T + 73iT^{2} \)
79 \( 1 + (-8.39 + 8.39i)T - 79iT^{2} \)
83 \( 1 + 12.7iT - 83T^{2} \)
89 \( 1 + 0.0162T + 89T^{2} \)
97 \( 1 + (2.58 + 2.58i)T + 97iT^{2} \)
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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

−12.41758950328427534050957150937, −11.58732660153130957745614381066, −10.32208138596411733441022672510, −9.678392750920799671885256860290, −9.095822481303207476802729818457, −7.63536003798852561756624830710, −6.22816882027152154973552723456, −4.54683002703047151160736198004, −3.14133259416675597224956970005, −2.55259228352894359481481975023, 1.97331883850138539965979356976, 3.42690233125573310360293875975, 5.47252615027671841506399497048, 6.82468284891202913857245089641, 7.37827009063694888566150066662, 8.094048389771458912080944512100, 9.389936919224786044679636202290, 10.41277608823336093401393332960, 12.10652257016228711630852915369, 12.66054064327173365833460409341

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