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Properties

Label 2-861-21.17-c1-0-76
Degree $2$
Conductor $861$
Sign $-0.427 + 0.903i$
Analytic cond. $6.87511$
Root an. cond. $2.62204$
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 + (−1.08 + 1.34i)3-s + (2.09 − 3.62i)4-s + (−1.86 − 3.23i)5-s + (0.664 − 4.25i)6-s + (1.06 − 2.42i)7-s + 5.44i·8-s + (−0.637 − 2.93i)9-s + (8.05 + 4.65i)10-s + (−1.36 − 0.785i)11-s + (2.61 + 6.76i)12-s + 2.25i·13-s + (0.718 + 6.54i)14-s + (6.39 + 0.998i)15-s + (−2.58 − 4.47i)16-s + (3.44 − 5.97i)17-s + ⋯
L(s)  = 1  + (−1.52 + 0.879i)2-s + (−0.627 + 0.778i)3-s + (1.04 − 1.81i)4-s + (−0.836 − 1.44i)5-s + (0.271 − 1.73i)6-s + (0.402 − 0.915i)7-s + 1.92i·8-s + (−0.212 − 0.977i)9-s + (2.54 + 1.47i)10-s + (−0.410 − 0.236i)11-s + (0.755 + 1.95i)12-s + 0.625i·13-s + (0.191 + 1.74i)14-s + (1.65 + 0.257i)15-s + (−0.646 − 1.11i)16-s + (0.836 − 1.44i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 861 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.427 + 0.903i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 861 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.427 + 0.903i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(861\)    =    \(3 \cdot 7 \cdot 41\)
Sign: $-0.427 + 0.903i$
Analytic conductor: \(6.87511\)
Root analytic conductor: \(2.62204\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{861} (206, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 861,\ (\ :1/2),\ -0.427 + 0.903i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.145277 - 0.229538i\)
\(L(\frac12)\) \(\approx\) \(0.145277 - 0.229538i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.08 - 1.34i)T \)
7 \( 1 + (-1.06 + 2.42i)T \)
41 \( 1 - T \)
good2 \( 1 + (2.15 - 1.24i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (1.86 + 3.23i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.36 + 0.785i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 2.25iT - 13T^{2} \)
17 \( 1 + (-3.44 + 5.97i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.08 + 0.629i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-5.97 + 3.44i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 7.15iT - 29T^{2} \)
31 \( 1 + (6.09 + 3.52i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.43 + 5.95i)T + (-18.5 + 32.0i)T^{2} \)
43 \( 1 + 4.61T + 43T^{2} \)
47 \( 1 + (4.01 + 6.95i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.90 - 2.25i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.56 - 4.43i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.77 + 2.75i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.40 + 5.90i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 7.12iT - 71T^{2} \)
73 \( 1 + (-2.63 - 1.52i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.18 - 12.4i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 10.4T + 83T^{2} \)
89 \( 1 + (-3.12 - 5.41i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 6.03iT - 97T^{2} \)
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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

−9.551374699960291469178328980731, −9.072415674979675177587142899748, −8.357008067851339115930688409270, −7.42454588114512613118739561878, −6.86498040551584523945652311611, −5.29259299657457780688366728288, −4.99820412014454156366208614682, −3.73666446499597185767176630978, −1.11657458121010770257407902400, −0.29496956763235805263433784042, 1.51936866266951325336468552392, 2.63080830687022347359562210458, 3.43315005884183411676769719366, 5.38985993962259865209273480253, 6.49562047368139579483412377398, 7.46369347648634939577578149930, 7.899494822718021975735874449500, 8.567598847232634169584254832242, 9.926973021993293436702538039659, 10.53813064119942548329009616219

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