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Properties

Label 2-1110-185.184-c1-0-18
Degree $2$
Conductor $1110$
Sign $0.952 - 0.304i$
Analytic cond. $8.86339$
Root an. cond. $2.97714$
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-s + i·3-s + 4-s + (−2.20 + 0.347i)5-s + i·6-s − 2.08i·7-s + 8-s − 9-s + (−2.20 + 0.347i)10-s + 6.52·11-s + i·12-s − 1.53·13-s − 2.08i·14-s + (−0.347 − 2.20i)15-s + 16-s + 5.28·17-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577i·3-s + 0.5·4-s + (−0.987 + 0.155i)5-s + 0.408i·6-s − 0.786i·7-s + 0.353·8-s − 0.333·9-s + (−0.698 + 0.109i)10-s + 1.96·11-s + 0.288i·12-s − 0.425·13-s − 0.556i·14-s + (−0.0897 − 0.570i)15-s + 0.250·16-s + 1.28·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 - 0.304i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.952 - 0.304i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.952 - 0.304i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (739, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ 0.952 - 0.304i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.371773828\)
\(L(\frac12)\) \(\approx\) \(2.371773828\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 - iT \)
5 \( 1 + (2.20 - 0.347i)T \)
37 \( 1 + (-0.927 - 6.01i)T \)
good7 \( 1 + 2.08iT - 7T^{2} \)
11 \( 1 - 6.52T + 11T^{2} \)
13 \( 1 + 1.53T + 13T^{2} \)
17 \( 1 - 5.28T + 17T^{2} \)
19 \( 1 + 4.07iT - 19T^{2} \)
23 \( 1 + 1.43T + 23T^{2} \)
29 \( 1 - 6.89iT - 29T^{2} \)
31 \( 1 - 8.19iT - 31T^{2} \)
41 \( 1 - 6.76T + 41T^{2} \)
43 \( 1 - 4.46T + 43T^{2} \)
47 \( 1 + 9.83iT - 47T^{2} \)
53 \( 1 + 6.51iT - 53T^{2} \)
59 \( 1 + 11.0iT - 59T^{2} \)
61 \( 1 + 4.91iT - 61T^{2} \)
67 \( 1 - 11.8iT - 67T^{2} \)
71 \( 1 + 6.07T + 71T^{2} \)
73 \( 1 - 13.6iT - 73T^{2} \)
79 \( 1 - 2.76iT - 79T^{2} \)
83 \( 1 + 15.8iT - 83T^{2} \)
89 \( 1 - 7.15iT - 89T^{2} \)
97 \( 1 + 19.0T + 97T^{2} \)
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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

−10.02559441291821841212922661591, −9.092799738692094641855249641033, −8.200997916034384069127924611019, −7.03810476668925779871633396957, −6.78483864044834802334717291422, −5.37788817285095998216403242599, −4.44636223716899947671936769397, −3.79027739461536638248252794851, −3.10912942311970962265354061575, −1.14341872044269555472647384517, 1.16680120506906394764470465326, 2.53920754529625145235605714417, 3.76915028278043640373517222038, 4.33606650501554542040763476391, 5.82087663685885145328435707804, 6.15351735866768755777604740705, 7.46957424912264921954095540693, 7.80796893185408305931586411250, 8.966137332895031632231740187490, 9.643581164879281643762611110271

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