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Properties

Label 2-231-21.20-c1-0-21
Degree $2$
Conductor $231$
Sign $-0.275 + 0.961i$
Analytic cond. $1.84454$
Root an. cond. $1.35814$
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  − 2.06i·2-s + (1.71 + 0.273i)3-s − 2.24·4-s + 1.49·5-s + (0.564 − 3.52i)6-s + (−2.62 − 0.319i)7-s + 0.510i·8-s + (2.85 + 0.936i)9-s − 3.08i·10-s + i·11-s + (−3.84 − 0.615i)12-s − 4.91i·13-s + (−0.657 + 5.41i)14-s + (2.56 + 0.410i)15-s − 3.44·16-s + 4.17·17-s + ⋯
L(s)  = 1  − 1.45i·2-s + (0.987 + 0.158i)3-s − 1.12·4-s + 0.670·5-s + (0.230 − 1.43i)6-s + (−0.992 − 0.120i)7-s + 0.180i·8-s + (0.950 + 0.312i)9-s − 0.976i·10-s + 0.301i·11-s + (−1.10 − 0.177i)12-s − 1.36i·13-s + (−0.175 + 1.44i)14-s + (0.661 + 0.105i)15-s − 0.860·16-s + 1.01·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.275 + 0.961i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.275 + 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $-0.275 + 0.961i$
Analytic conductor: \(1.84454\)
Root analytic conductor: \(1.35814\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{231} (188, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 231,\ (\ :1/2),\ -0.275 + 0.961i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.00025 - 1.32785i\)
\(L(\frac12)\) \(\approx\) \(1.00025 - 1.32785i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.71 - 0.273i)T \)
7 \( 1 + (2.62 + 0.319i)T \)
11 \( 1 - iT \)
good2 \( 1 + 2.06iT - 2T^{2} \)
5 \( 1 - 1.49T + 5T^{2} \)
13 \( 1 + 4.91iT - 13T^{2} \)
17 \( 1 - 4.17T + 17T^{2} \)
19 \( 1 - 5.12iT - 19T^{2} \)
23 \( 1 - 1.67iT - 23T^{2} \)
29 \( 1 - 1.88iT - 29T^{2} \)
31 \( 1 - 10.2iT - 31T^{2} \)
37 \( 1 + 9.67T + 37T^{2} \)
41 \( 1 + 2.22T + 41T^{2} \)
43 \( 1 + 1.70T + 43T^{2} \)
47 \( 1 - 4.74T + 47T^{2} \)
53 \( 1 + 3.56iT - 53T^{2} \)
59 \( 1 - 7.28T + 59T^{2} \)
61 \( 1 + 5.73iT - 61T^{2} \)
67 \( 1 - 7.64T + 67T^{2} \)
71 \( 1 + 11.0iT - 71T^{2} \)
73 \( 1 + 3.21iT - 73T^{2} \)
79 \( 1 + 6.24T + 79T^{2} \)
83 \( 1 + 13.4T + 83T^{2} \)
89 \( 1 + 0.928T + 89T^{2} \)
97 \( 1 - 9.39iT - 97T^{2} \)
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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

−12.26738694289878708136478564902, −10.45668420662495310328047903316, −10.16425028297423676737917041864, −9.469405848879033457040068475505, −8.326611751954123847789702732765, −7.00485751489817756647613163405, −5.41107598261823837005018884053, −3.64059005459868820705906720366, −3.06101543925464221148009096667, −1.62961011676083936963200269316, 2.42110668164957900450886721199, 4.07265542324444273042924766098, 5.62944586241116690812407434597, 6.61937870113872909916659036004, 7.28623964805456564579012037392, 8.510420409754231657913790550440, 9.263592941987138660875971118662, 9.956773814567249350075462226125, 11.69348607973133180367739557967, 13.03589670251812224562351367061

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