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Properties

Label 2-231-231.65-c1-0-15
Degree $2$
Conductor $231$
Sign $0.926 + 0.377i$
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 + (1.04 + 1.37i)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 + (−0.798 + 2.89i)9-s + (−1.82 + 1.05i)10-s + (−3.17 + 0.964i)11-s + (0.351 − 0.839i)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 + (0.605 + 0.795i)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 + (−0.266 + 0.963i)9-s + (−0.576 + 0.333i)10-s + (−0.956 + 0.290i)11-s + (0.101 − 0.242i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.926 + 0.377i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.926 + 0.377i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $0.926 + 0.377i$
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: \((2,\ 231,\ (\ :1/2),\ 0.926 + 0.377i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.91589 - 0.375193i\)
\(L(\frac12)\) \(\approx\) \(1.91589 - 0.375193i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.04 - 1.37i)T \)
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

−12.06377372550138006779873226014, −11.05583434008915981786400552631, −10.64209452206507650980784909847, −9.366211617224770254315859035944, −8.188499920442566538780616816183, −7.58169344560583686362271564258, −5.15810842755909097106194055923, −4.62413946543658500127249805588, −3.31580072287920880801290387363, −2.22503832917085154124080513737, 1.90554761591351372296917865559, 3.81099443756697979721307900205, 5.05762472723726379070253307882, 6.37225960461013236083364932196, 7.25682843106846414875848128144, 7.934834553872229144750013408187, 8.810078234601007671312426754126, 10.58517568979879242692485289276, 11.29971882357897397462462629220, 12.61078300946073108167775488029

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