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Properties

Label 2-231-11.5-c1-0-6
Degree $2$
Conductor $231$
Sign $0.785 - 0.618i$
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  + (0.557 + 1.71i)2-s + (−0.809 − 0.587i)3-s + (−1.01 + 0.740i)4-s + (0.858 − 2.64i)5-s + (0.557 − 1.71i)6-s + (0.809 − 0.587i)7-s + (1.08 + 0.785i)8-s + (0.309 + 0.951i)9-s + 5.01·10-s + (3.22 + 0.754i)11-s + 1.25·12-s + (0.714 + 2.19i)13-s + (1.46 + 1.06i)14-s + (−2.24 + 1.63i)15-s + (−1.52 + 4.69i)16-s + (0.822 − 2.53i)17-s + ⋯
L(s)  = 1  + (0.394 + 1.21i)2-s + (−0.467 − 0.339i)3-s + (−0.509 + 0.370i)4-s + (0.383 − 1.18i)5-s + (0.227 − 0.700i)6-s + (0.305 − 0.222i)7-s + (0.382 + 0.277i)8-s + (0.103 + 0.317i)9-s + 1.58·10-s + (0.973 + 0.227i)11-s + 0.363·12-s + (0.198 + 0.610i)13-s + (0.390 + 0.283i)14-s + (−0.580 + 0.421i)15-s + (−0.381 + 1.17i)16-s + (0.199 − 0.613i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.785 - 0.618i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.785 - 0.618i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $0.785 - 0.618i$
Analytic conductor: \(1.84454\)
Root analytic conductor: \(1.35814\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{231} (148, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 231,\ (\ :1/2),\ 0.785 - 0.618i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.43344 + 0.496883i\)
\(L(\frac12)\) \(\approx\) \(1.43344 + 0.496883i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.809 + 0.587i)T \)
7 \( 1 + (-0.809 + 0.587i)T \)
11 \( 1 + (-3.22 - 0.754i)T \)
good2 \( 1 + (-0.557 - 1.71i)T + (-1.61 + 1.17i)T^{2} \)
5 \( 1 + (-0.858 + 2.64i)T + (-4.04 - 2.93i)T^{2} \)
13 \( 1 + (-0.714 - 2.19i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-0.822 + 2.53i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (6.53 + 4.75i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 2.43T + 23T^{2} \)
29 \( 1 + (-6.11 + 4.44i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-2.85 - 8.79i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (9.05 - 6.58i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-0.242 - 0.176i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 7.29T + 43T^{2} \)
47 \( 1 + (0.370 + 0.269i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (1.60 + 4.93i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (9.93 - 7.21i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (0.561 - 1.72i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 1.42T + 67T^{2} \)
71 \( 1 + (-0.172 + 0.530i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-7.81 + 5.67i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (3.18 + 9.81i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (2.62 - 8.09i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 1.56T + 89T^{2} \)
97 \( 1 + (0.164 + 0.507i)T + (-78.4 + 57.0i)T^{2} \)
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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

−12.41559992807776457341378148729, −11.61832634607991668250474541587, −10.36826313216196092629273896315, −8.949387227120922250490099718228, −8.290961367097154531831774797346, −6.87285024935964935122443565313, −6.36939787700034907813813855707, −4.99945008693105216758662649642, −4.50089150397016223699318429441, −1.58130244778678269397917884719, 1.87691376263516869946749327048, 3.29324896560485095460291470989, 4.25163088147272008265352283088, 5.88043046316493493055015930081, 6.74902144123799718862464621249, 8.293037431330149754388047889890, 9.779183542613736937753830227421, 10.56243798793453170252627772050, 10.97323315778137781510523277598, 12.02521221987906803471523175662

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