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Properties

Label 2-1110-111.80-c1-0-11
Degree $2$
Conductor $1110$
Sign $0.888 + 0.458i$
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  + (−0.707 − 0.707i)2-s + (−1.08 − 1.35i)3-s + 1.00i·4-s + (−0.707 + 0.707i)5-s + (−0.189 + 1.72i)6-s − 3.77·7-s + (0.707 − 0.707i)8-s + (−0.651 + 2.92i)9-s + 1.00·10-s + 1.81·11-s + (1.35 − 1.08i)12-s + (−3.90 − 3.90i)13-s + (2.67 + 2.67i)14-s + (1.72 + 0.189i)15-s − 1.00·16-s + (−1.04 + 1.04i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (−0.625 − 0.780i)3-s + 0.500i·4-s + (−0.316 + 0.316i)5-s + (−0.0772 + 0.702i)6-s − 1.42·7-s + (0.250 − 0.250i)8-s + (−0.217 + 0.976i)9-s + 0.316·10-s + 0.547·11-s + (0.390 − 0.312i)12-s + (−1.08 − 1.08i)13-s + (0.713 + 0.713i)14-s + (0.444 + 0.0488i)15-s − 0.250·16-s + (−0.253 + 0.253i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.888 + 0.458i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.888 + 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.888 + 0.458i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ 0.888 + 0.458i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5215866783\)
\(L(\frac12)\) \(\approx\) \(0.5215866783\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 0.707i)T \)
3 \( 1 + (1.08 + 1.35i)T \)
5 \( 1 + (0.707 - 0.707i)T \)
37 \( 1 + (-5.83 - 1.70i)T \)
good7 \( 1 + 3.77T + 7T^{2} \)
11 \( 1 - 1.81T + 11T^{2} \)
13 \( 1 + (3.90 + 3.90i)T + 13iT^{2} \)
17 \( 1 + (1.04 - 1.04i)T - 17iT^{2} \)
19 \( 1 + (1.01 + 1.01i)T + 19iT^{2} \)
23 \( 1 + (5.79 - 5.79i)T - 23iT^{2} \)
29 \( 1 + (-3.80 - 3.80i)T + 29iT^{2} \)
31 \( 1 + (-4.46 + 4.46i)T - 31iT^{2} \)
41 \( 1 - 9.74T + 41T^{2} \)
43 \( 1 + (-6.17 - 6.17i)T + 43iT^{2} \)
47 \( 1 + 7.59iT - 47T^{2} \)
53 \( 1 - 1.46iT - 53T^{2} \)
59 \( 1 + (3.27 - 3.27i)T - 59iT^{2} \)
61 \( 1 + (0.251 - 0.251i)T - 61iT^{2} \)
67 \( 1 - 1.79iT - 67T^{2} \)
71 \( 1 + 14.5iT - 71T^{2} \)
73 \( 1 - 7.13iT - 73T^{2} \)
79 \( 1 + (8.05 + 8.05i)T + 79iT^{2} \)
83 \( 1 + 7.15iT - 83T^{2} \)
89 \( 1 + (-7.47 - 7.47i)T + 89iT^{2} \)
97 \( 1 + (-3.22 - 3.22i)T + 97iT^{2} \)
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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

−9.950960006772359441971378972845, −9.130532737631717153135556020184, −7.897497518519583536107988392957, −7.43509735410243904931151954690, −6.44569650475884589182653658226, −5.86539538574347137576636203855, −4.44107316269192875231461067352, −3.21138047802018085140935863539, −2.35896770737372586751187739768, −0.69517825042504170182787629448, 0.49104834437056938165970715956, 2.61430573385554494326747569346, 4.10629979070801145861436681402, 4.57339024145790675075902316000, 5.96883553646648031825283449207, 6.43561364756473475376761680569, 7.23120839225047863014109813329, 8.470476888202055804330451702002, 9.329825238603849942160894934662, 9.711571802651389488355546891142

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