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Properties

Label 2-231-11.5-c1-0-7
Degree $2$
Conductor $231$
Sign $0.569 - 0.821i$
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  + (0.449 + 1.38i)2-s + (0.809 + 0.587i)3-s + (−0.0967 + 0.0703i)4-s + (0.737 − 2.27i)5-s + (−0.449 + 1.38i)6-s + (0.809 − 0.587i)7-s + (2.21 + 1.60i)8-s + (0.309 + 0.951i)9-s + 3.47·10-s + (−3.30 + 0.224i)11-s − 0.119·12-s + (−0.281 − 0.867i)13-s + (1.17 + 0.855i)14-s + (1.93 − 1.40i)15-s + (−1.30 + 4.01i)16-s + (−0.675 + 2.07i)17-s + ⋯
L(s)  = 1  + (0.318 + 0.979i)2-s + (0.467 + 0.339i)3-s + (−0.0483 + 0.0351i)4-s + (0.329 − 1.01i)5-s + (−0.183 + 0.565i)6-s + (0.305 − 0.222i)7-s + (0.783 + 0.568i)8-s + (0.103 + 0.317i)9-s + 1.09·10-s + (−0.997 + 0.0676i)11-s − 0.0345·12-s + (−0.0781 − 0.240i)13-s + (0.314 + 0.228i)14-s + (0.498 − 0.362i)15-s + (−0.326 + 1.00i)16-s + (−0.163 + 0.504i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.569 - 0.821i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.569 - 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $0.569 - 0.821i$
Analytic conductor: \(1.84454\)
Root analytic conductor: \(1.35814\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{231} (148, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 231,\ (\ :1/2),\ 0.569 - 0.821i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.63014 + 0.853299i\)
\(L(\frac12)\) \(\approx\) \(1.63014 + 0.853299i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.809 - 0.587i)T \)
7 \( 1 + (-0.809 + 0.587i)T \)
11 \( 1 + (3.30 - 0.224i)T \)
good2 \( 1 + (-0.449 - 1.38i)T + (-1.61 + 1.17i)T^{2} \)
5 \( 1 + (-0.737 + 2.27i)T + (-4.04 - 2.93i)T^{2} \)
13 \( 1 + (0.281 + 0.867i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (0.675 - 2.07i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (0.575 + 0.418i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 7.80T + 23T^{2} \)
29 \( 1 + (7.17 - 5.21i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (2.04 + 6.30i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-3.40 + 2.47i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-9.67 - 7.02i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 7.51T + 43T^{2} \)
47 \( 1 + (5.07 + 3.68i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (2.02 + 6.24i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-7.78 + 5.65i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (0.0193 - 0.0596i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 11.3T + 67T^{2} \)
71 \( 1 + (1.84 - 5.67i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-3.54 + 2.57i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (1.50 + 4.61i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-3.10 + 9.55i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 0.261T + 89T^{2} \)
97 \( 1 + (-3.05 - 9.40i)T + (-78.4 + 57.0i)T^{2} \)
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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

−12.86411333348462029158789935206, −11.26584884373353845130340240292, −10.33373439290330746284296586586, −9.250232080370887109149590744811, −8.111026646096794219317264861712, −7.57840133793551966974797299162, −5.99146375651087857172067772424, −5.19040582460461893143084517402, −4.17908474513799843047751550288, −2.03895737164432998325077806206, 2.10385443391859311097296875415, 2.84608626973247902184466851525, 4.18187573610993624659485660980, 5.87886460490473163317438962746, 7.18133027523369205282680099048, 7.921551426128284886871589003040, 9.434848281434137180755968569989, 10.41384779422841808661843905059, 11.08286120541542916712983014787, 12.04379358814174241516588059545

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