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Properties

Label 2-231-231.32-c1-0-26
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: $\chi_{231} (32, \cdot )$
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

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

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