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Properties

Label 2-231-231.65-c1-0-4
Degree $2$
Conductor $231$
Sign $-0.361 - 0.932i$
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  + (−0.724 + 1.57i)3-s + (1 + 1.73i)4-s + (1.22 + 0.707i)5-s + (−1.32 + 2.29i)7-s + (−1.94 − 2.28i)9-s + (−2.23 − 2.45i)11-s + (−3.44 + 0.317i)12-s − 4.58i·13-s + (−2 + 1.41i)15-s + (−1.99 + 3.46i)16-s + (3.24 + 5.61i)17-s + (3.96 + 2.29i)19-s + 2.82i·20-s + (−2.64 − 3.74i)21-s + (4.89 + 2.82i)23-s + ⋯
L(s)  = 1  + (−0.418 + 0.908i)3-s + (0.5 + 0.866i)4-s + (0.547 + 0.316i)5-s + (−0.499 + 0.866i)7-s + (−0.649 − 0.760i)9-s + (−0.673 − 0.739i)11-s + (−0.995 + 0.0917i)12-s − 1.27i·13-s + (−0.516 + 0.365i)15-s + (−0.499 + 0.866i)16-s + (0.785 + 1.36i)17-s + (0.910 + 0.525i)19-s + 0.632i·20-s + (−0.577 − 0.816i)21-s + (1.02 + 0.589i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.361 - 0.932i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.361 - 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $-0.361 - 0.932i$
Analytic conductor: \(1.84454\)
Root analytic conductor: \(1.35814\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{231} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 231,\ (\ :1/2),\ -0.361 - 0.932i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.658835 + 0.962304i\)
\(L(\frac12)\) \(\approx\) \(0.658835 + 0.962304i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.724 - 1.57i)T \)
7 \( 1 + (1.32 - 2.29i)T \)
11 \( 1 + (2.23 + 2.45i)T \)
good2 \( 1 + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-1.22 - 0.707i)T + (2.5 + 4.33i)T^{2} \)
13 \( 1 + 4.58iT - 13T^{2} \)
17 \( 1 + (-3.24 - 5.61i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.96 - 2.29i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.89 - 2.82i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 6.48T + 29T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.5 + 6.06i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 6.48T + 41T^{2} \)
43 \( 1 - 4.58iT - 43T^{2} \)
47 \( 1 + (-4.89 - 2.82i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.12 + 3.53i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.44 + 1.41i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 1.41iT - 71T^{2} \)
73 \( 1 + (-3.96 + 2.29i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (11.9 + 6.87i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 6.48T + 83T^{2} \)
89 \( 1 + (13.4 + 7.77i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 8T + 97T^{2} \)
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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

−12.51080396645180299503090816882, −11.40919954999339950099585762961, −10.61708027024591272142848938123, −9.744299993001432158999978406089, −8.621076452068099939007246988743, −7.63366526698823821351853128666, −5.92583224387280410624165033613, −5.64303073082398837935849591832, −3.59911000572291674731175868070, −2.81557152047727657099104124170, 1.07044999052559405569571913755, 2.51602633307476068229502284736, 4.86611491518694382376724148939, 5.75440118767340150523223288119, 7.11576337552658518501187350134, 7.24152718765596055314514436027, 9.273175875536152219373241977702, 9.949216849194931739907744714088, 11.08167440383062745962308220875, 11.79197278753085534568943015713

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