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Properties

Label 2-231-77.54-c1-0-4
Degree $2$
Conductor $231$
Sign $-0.206 - 0.978i$
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  + (2.23 + 1.29i)2-s + (−0.866 + 0.5i)3-s + (2.34 + 4.06i)4-s + (−1.30 − 0.751i)5-s − 2.58·6-s + (−1.55 + 2.13i)7-s + 6.95i·8-s + (0.499 − 0.866i)9-s + (−1.94 − 3.36i)10-s + (3.00 − 1.41i)11-s + (−4.06 − 2.34i)12-s + 3.99·13-s + (−6.25 + 2.77i)14-s + 1.50·15-s + (−4.30 + 7.46i)16-s + (−1.38 − 2.39i)17-s + ⋯
L(s)  = 1  + (1.58 + 0.914i)2-s + (−0.499 + 0.288i)3-s + (1.17 + 2.03i)4-s + (−0.581 − 0.335i)5-s − 1.05·6-s + (−0.588 + 0.808i)7-s + 2.45i·8-s + (0.166 − 0.288i)9-s + (−0.614 − 1.06i)10-s + (0.904 − 0.426i)11-s + (−1.17 − 0.676i)12-s + 1.10·13-s + (−1.67 + 0.742i)14-s + 0.387·15-s + (−1.07 + 1.86i)16-s + (−0.335 − 0.580i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.206 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.206 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $-0.206 - 0.978i$
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: \((2,\ 231,\ (\ :1/2),\ -0.206 - 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.40433 + 1.73197i\)
\(L(\frac12)\) \(\approx\) \(1.40433 + 1.73197i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.55 - 2.13i)T \)
11 \( 1 + (-3.00 + 1.41i)T \)
good2 \( 1 + (-2.23 - 1.29i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (1.30 + 0.751i)T + (2.5 + 4.33i)T^{2} \)
13 \( 1 - 3.99T + 13T^{2} \)
17 \( 1 + (1.38 + 2.39i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.45 + 4.25i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.0359 + 0.0623i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 5.06iT - 29T^{2} \)
31 \( 1 + (8.84 - 5.10i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.90 - 6.76i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 2.98T + 41T^{2} \)
43 \( 1 - 5.41iT - 43T^{2} \)
47 \( 1 + (-4.48 - 2.59i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (6.23 + 10.7i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.52 - 3.18i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.688 - 1.19i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.27 - 3.94i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 9.59T + 71T^{2} \)
73 \( 1 + (-4.48 - 7.76i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (10.0 + 5.82i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 9.34T + 83T^{2} \)
89 \( 1 + (11.7 + 6.80i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 2.63iT - 97T^{2} \)
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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

−12.58726162776133282869167848139, −11.74044360884922491477188018314, −11.24157112059414135444467760820, −9.303178805810319414208600799998, −8.319687452656147956785077534965, −6.91954475094359235169703441559, −6.18237788064048277265684694119, −5.25295436773716578111865661914, −4.15570892465786228851124260901, −3.17464406094620992631685608090, 1.56506715178875908329284687374, 3.62619537076472660362428257167, 3.97643540916810791998319286888, 5.58553729298741184131201074014, 6.48561070014796592424825665730, 7.43261467114510443387824841626, 9.429156036330187916408294656270, 10.81237107690655060055822128213, 10.97361668034363227006795021555, 12.16645721121624147106669044840

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