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Properties

Label 2-189-21.5-c1-0-7
Degree $2$
Conductor $189$
Sign $0.528 - 0.849i$
Analytic cond. $1.50917$
Root an. cond. $1.22848$
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  + (2.15 + 1.24i)2-s + (2.10 + 3.64i)4-s + (0.617 − 1.07i)5-s + (−1.53 − 2.15i)7-s + 5.49i·8-s + (2.66 − 1.53i)10-s + (−4.75 + 2.74i)11-s − 2.96i·13-s + (−0.617 − 6.56i)14-s + (−2.63 + 4.56i)16-s + (2.15 + 3.73i)17-s + (−4.80 − 2.77i)19-s + 5.19·20-s − 13.6·22-s + (1.41 + 0.815i)23-s + ⋯
L(s)  = 1  + (1.52 + 0.880i)2-s + (1.05 + 1.82i)4-s + (0.276 − 0.478i)5-s + (−0.579 − 0.815i)7-s + 1.94i·8-s + (0.843 − 0.486i)10-s + (−1.43 + 0.827i)11-s − 0.823i·13-s + (−0.165 − 1.75i)14-s + (−0.658 + 1.14i)16-s + (0.523 + 0.906i)17-s + (−1.10 − 0.636i)19-s + 1.16·20-s − 2.91·22-s + (0.294 + 0.170i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.528 - 0.849i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.528 - 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.528 - 0.849i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ 0.528 - 0.849i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.08378 + 1.15787i\)
\(L(\frac12)\) \(\approx\) \(2.08378 + 1.15787i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (1.53 + 2.15i)T \)
good2 \( 1 + (-2.15 - 1.24i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (-0.617 + 1.07i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (4.75 - 2.74i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 2.96iT - 13T^{2} \)
17 \( 1 + (-2.15 - 3.73i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.80 + 2.77i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.41 - 0.815i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 0.509iT - 29T^{2} \)
31 \( 1 + (-7.04 + 4.06i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.53 + 2.65i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 0.354T + 41T^{2} \)
43 \( 1 + 0.0637T + 43T^{2} \)
47 \( 1 + (4.93 - 8.54i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.57 + 2.06i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.53 + 2.66i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.97 + 3.44i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.16 - 10.6i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 4.63iT - 71T^{2} \)
73 \( 1 + (6.09 - 3.51i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.165 + 0.287i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 7.39T + 83T^{2} \)
89 \( 1 + (-1.71 + 2.97i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 4.45iT - 97T^{2} \)
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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

−13.01301223653950546263060129953, −12.49052014523669546212731516363, −10.87907689867098247732724505491, −9.902268131171362863326715351871, −8.165099704711540754163925334117, −7.35036129908188750761680330442, −6.24264715832649859276393992437, −5.20685132617632817966753998634, −4.27659345897763331557502614994, −2.87892996628006335990053182938, 2.40723375473541433623573699468, 3.17840961417834810906292785729, 4.76448536684778627589160675360, 5.79514375959354658736634987734, 6.63088350349991418615490899715, 8.461335570864760484515702999687, 9.977714596405673132722621053665, 10.66882575015911367037752186137, 11.70793214797842799344632655587, 12.44314167826670650916993670752

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