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Properties

Label 2-231-231.32-c1-0-25
Degree $2$
Conductor $231$
Sign $-0.801 + 0.597i$
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  + (−0.794 − 1.37i)2-s + (0.668 − 1.59i)3-s + (−0.262 + 0.454i)4-s + (1.14 − 0.662i)5-s + (−2.73 + 0.348i)6-s + (2.48 − 0.904i)7-s − 2.34·8-s + (−2.10 − 2.13i)9-s + (−1.82 − 1.05i)10-s + (3.17 + 0.964i)11-s + (0.550 + 0.723i)12-s + 3.14i·13-s + (−3.22 − 2.70i)14-s + (−0.290 − 2.27i)15-s + (2.38 + 4.13i)16-s + (1.28 − 2.22i)17-s + ⋯
L(s)  = 1  + (−0.561 − 0.973i)2-s + (0.386 − 0.922i)3-s + (−0.131 + 0.227i)4-s + (0.513 − 0.296i)5-s + (−1.11 + 0.142i)6-s + (0.939 − 0.341i)7-s − 0.828·8-s + (−0.701 − 0.712i)9-s + (−0.576 − 0.333i)10-s + (0.956 + 0.290i)11-s + (0.159 + 0.208i)12-s + 0.870i·13-s + (−0.860 − 0.722i)14-s + (−0.0751 − 0.588i)15-s + (0.596 + 1.03i)16-s + (0.312 − 0.540i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.801 + 0.597i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.801 + 0.597i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $-0.801 + 0.597i$
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: \((2,\ 231,\ (\ :1/2),\ -0.801 + 0.597i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.379659 - 1.14394i\)
\(L(\frac12)\) \(\approx\) \(0.379659 - 1.14394i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.668 + 1.59i)T \)
7 \( 1 + (-2.48 + 0.904i)T \)
11 \( 1 + (-3.17 - 0.964i)T \)
good2 \( 1 + (0.794 + 1.37i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-1.14 + 0.662i)T + (2.5 - 4.33i)T^{2} \)
13 \( 1 - 3.14iT - 13T^{2} \)
17 \( 1 + (-1.28 + 2.22i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.54 - 2.04i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (6.23 - 3.59i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 4.92T + 29T^{2} \)
31 \( 1 + (0.543 - 0.940i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.551 + 0.955i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 6.61T + 41T^{2} \)
43 \( 1 + 5.86iT - 43T^{2} \)
47 \( 1 + (-1.18 + 0.685i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.25 + 1.30i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-9.38 - 5.41i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (8.35 - 4.82i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.67 + 8.09i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 11.4iT - 71T^{2} \)
73 \( 1 + (-11.3 - 6.55i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.91 + 3.41i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 13.1T + 83T^{2} \)
89 \( 1 + (11.1 - 6.41i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 6.94T + 97T^{2} \)
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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

−11.88694346094970259188217240779, −11.00514473037561877455892171453, −9.763704931870613168796130283748, −9.056362149316830005745899942436, −8.101382799209657737627788592630, −6.86477717587836700704262815016, −5.75431587236976806849083834034, −3.93443084091230414481040072424, −2.13386571201167487806953424687, −1.36437940667947119182795905818, 2.58223191907342828588325485865, 4.17988408363677547302232785095, 5.62566707129051265879219760931, 6.44275328093347621672218401438, 8.066304388215062828386545634590, 8.406045812734480090091401333873, 9.473650990515969107254418192758, 10.39651556024974700448444914711, 11.40794201617515314930255820774, 12.50814443218623160623918059268

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