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Properties

Label 2-231-77.76-c1-0-10
Degree $2$
Conductor $231$
Sign $-0.563 + 0.825i$
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  − 2.08i·2-s + i·3-s − 2.34·4-s − 0.833i·5-s + 2.08·6-s + (2.19 − 1.47i)7-s + 0.726i·8-s − 9-s − 1.73·10-s + (−1.23 − 3.07i)11-s − 2.34i·12-s + 4.54·13-s + (−3.06 − 4.58i)14-s + 0.833·15-s − 3.18·16-s − 6.38·17-s + ⋯
L(s)  = 1  − 1.47i·2-s + 0.577i·3-s − 1.17·4-s − 0.372i·5-s + 0.851·6-s + (0.830 − 0.556i)7-s + 0.256i·8-s − 0.333·9-s − 0.549·10-s + (−0.372 − 0.927i)11-s − 0.677i·12-s + 1.26·13-s + (−0.820 − 1.22i)14-s + 0.215·15-s − 0.795·16-s − 1.54·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.563 + 0.825i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.563 + 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $-0.563 + 0.825i$
Analytic conductor: \(1.84454\)
Root analytic conductor: \(1.35814\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{231} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 231,\ (\ :1/2),\ -0.563 + 0.825i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.582742 - 1.10328i\)
\(L(\frac12)\) \(\approx\) \(0.582742 - 1.10328i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - iT \)
7 \( 1 + (-2.19 + 1.47i)T \)
11 \( 1 + (1.23 + 3.07i)T \)
good2 \( 1 + 2.08iT - 2T^{2} \)
5 \( 1 + 0.833iT - 5T^{2} \)
13 \( 1 - 4.54T + 13T^{2} \)
17 \( 1 + 6.38T + 17T^{2} \)
19 \( 1 + 2.56T + 19T^{2} \)
23 \( 1 - 9.16T + 23T^{2} \)
29 \( 1 - 0.152iT - 29T^{2} \)
31 \( 1 - 8.36iT - 31T^{2} \)
37 \( 1 + 5.30T + 37T^{2} \)
41 \( 1 - 3.21T + 41T^{2} \)
43 \( 1 - 5.66iT - 43T^{2} \)
47 \( 1 - 6.00iT - 47T^{2} \)
53 \( 1 - 4.69T + 53T^{2} \)
59 \( 1 - 1.05iT - 59T^{2} \)
61 \( 1 - 8.34T + 61T^{2} \)
67 \( 1 - 4.33T + 67T^{2} \)
71 \( 1 + 4.80T + 71T^{2} \)
73 \( 1 + 10.6T + 73T^{2} \)
79 \( 1 - 5.70iT - 79T^{2} \)
83 \( 1 + 10.7T + 83T^{2} \)
89 \( 1 - 8.13iT - 89T^{2} \)
97 \( 1 + 17.7iT - 97T^{2} \)
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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

−11.36033461747671311354521266274, −10.98470022067449572760846938329, −10.46854345084064819836714043079, −8.838745534067738303841437332959, −8.678989970901381469120827748526, −6.75302318224998455803491544502, −5.05216271017575180761675206732, −4.13599718550586049303599729058, −2.95285182693675577226824828217, −1.18362106346580001229351187779, 2.25403187592937118343221357812, 4.52586587242896503555783298353, 5.58760786551812287730479251766, 6.68063370421664983686946136064, 7.28703384426679142636298159537, 8.499704276329437291396254881240, 8.920926105371560711797081277966, 10.80203199797861047551074364637, 11.47871236916849891660061891462, 12.93595170159613246835686372469

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