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Properties

Label 2-231-77.54-c1-0-14
Degree $2$
Conductor $231$
Sign $-0.869 + 0.494i$
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  + (−0.567 − 0.327i)2-s + (0.866 − 0.5i)3-s + (−0.785 − 1.36i)4-s + (−2.94 − 1.70i)5-s − 0.655·6-s + (2.64 − 0.162i)7-s + 2.33i·8-s + (0.499 − 0.866i)9-s + (1.11 + 1.92i)10-s + (−3.31 + 0.193i)11-s + (−1.36 − 0.785i)12-s − 5.44·13-s + (−1.55 − 0.772i)14-s − 3.40·15-s + (−0.804 + 1.39i)16-s + (−0.850 − 1.47i)17-s + ⋯
L(s)  = 1  + (−0.401 − 0.231i)2-s + (0.499 − 0.288i)3-s + (−0.392 − 0.680i)4-s + (−1.31 − 0.760i)5-s − 0.267·6-s + (0.998 − 0.0615i)7-s + 0.827i·8-s + (0.166 − 0.288i)9-s + (0.352 + 0.610i)10-s + (−0.998 + 0.0583i)11-s + (−0.392 − 0.226i)12-s − 1.51·13-s + (−0.414 − 0.206i)14-s − 0.878·15-s + (−0.201 + 0.348i)16-s + (−0.206 − 0.357i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.869 + 0.494i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.869 + 0.494i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $-0.869 + 0.494i$
Analytic conductor: \(1.84454\)
Root analytic conductor: \(1.35814\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{231} (208, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 231,\ (\ :1/2),\ -0.869 + 0.494i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.175367 - 0.663402i\)
\(L(\frac12)\) \(\approx\) \(0.175367 - 0.663402i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 + (-2.64 + 0.162i)T \)
11 \( 1 + (3.31 - 0.193i)T \)
good2 \( 1 + (0.567 + 0.327i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (2.94 + 1.70i)T + (2.5 + 4.33i)T^{2} \)
13 \( 1 + 5.44T + 13T^{2} \)
17 \( 1 + (0.850 + 1.47i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.12 + 5.41i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.09 + 5.35i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 3.61iT - 29T^{2} \)
31 \( 1 + (-5.63 + 3.25i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.0828 - 0.143i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 1.35T + 41T^{2} \)
43 \( 1 - 3.12iT - 43T^{2} \)
47 \( 1 + (0.693 + 0.400i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.20 + 5.54i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.54 + 0.892i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.78 + 3.08i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.44 - 7.70i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.59T + 71T^{2} \)
73 \( 1 + (5.41 + 9.37i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.09 + 2.36i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 14.2T + 83T^{2} \)
89 \( 1 + (-3.31 - 1.91i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 3.87iT - 97T^{2} \)
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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

−11.70535155944519539120233172497, −10.92009714435984158057469211829, −9.727152982973656608830540748071, −8.718825613012048845474365999771, −8.022795728702360145670124061835, −7.20406922580538361910612806754, −4.95349261824842586167354138698, −4.70909656632688379278801538501, −2.53682975567795601048535015970, −0.61469343413747766012373319418, 2.83748464960432916395631428114, 3.95927436550831105421411131765, 5.04455465887250207088147501613, 7.24703374715760351693139326175, 7.77107384365302738095519284760, 8.288007693578825006270773434566, 9.633321426992756267675745374843, 10.60012738004572672414722792108, 11.71454055593133246378723246275, 12.36313066640614982684431397954

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