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Properties

Label 2-2034-113.112-c1-0-28
Degree $2$
Conductor $2034$
Sign $0.881 - 0.471i$
Analytic cond. $16.2415$
Root an. cond. $4.03008$
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 + 4-s + 1.63i·5-s + 3.55·7-s + 8-s + 1.63i·10-s + 3.33·11-s − 1.82·13-s + 3.55·14-s + 16-s − 0.106i·17-s + 4.06i·19-s + 1.63i·20-s + 3.33·22-s − 8.58i·23-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + 0.733i·5-s + 1.34·7-s + 0.353·8-s + 0.518i·10-s + 1.00·11-s − 0.506·13-s + 0.950·14-s + 0.250·16-s − 0.0258i·17-s + 0.932i·19-s + 0.366i·20-s + 0.710·22-s − 1.78i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 - 0.471i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.881 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2034\)    =    \(2 \cdot 3^{2} \cdot 113\)
Sign: $0.881 - 0.471i$
Analytic conductor: \(16.2415\)
Root analytic conductor: \(4.03008\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2034} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2034,\ (\ :1/2),\ 0.881 - 0.471i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.442682902\)
\(L(\frac12)\) \(\approx\) \(3.442682902\)
\(L(\frac{3}{2})\) 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 \( 1 \)
113 \( 1 + (9.37 - 5.01i)T \)
good5 \( 1 - 1.63iT - 5T^{2} \)
7 \( 1 - 3.55T + 7T^{2} \)
11 \( 1 - 3.33T + 11T^{2} \)
13 \( 1 + 1.82T + 13T^{2} \)
17 \( 1 + 0.106iT - 17T^{2} \)
19 \( 1 - 4.06iT - 19T^{2} \)
23 \( 1 + 8.58iT - 23T^{2} \)
29 \( 1 + 3.75iT - 29T^{2} \)
31 \( 1 - 4.64T + 31T^{2} \)
37 \( 1 - 4.24iT - 37T^{2} \)
41 \( 1 + 7.62T + 41T^{2} \)
43 \( 1 - 11.8iT - 43T^{2} \)
47 \( 1 - 3.97iT - 47T^{2} \)
53 \( 1 + 7.55T + 53T^{2} \)
59 \( 1 - 3.60iT - 59T^{2} \)
61 \( 1 + 1.82T + 61T^{2} \)
67 \( 1 + 10.3iT - 67T^{2} \)
71 \( 1 + 2.25iT - 71T^{2} \)
73 \( 1 + 14.1iT - 73T^{2} \)
79 \( 1 - 4.19iT - 79T^{2} \)
83 \( 1 - 8.51T + 83T^{2} \)
89 \( 1 + 10.6iT - 89T^{2} \)
97 \( 1 - 17.1T + 97T^{2} \)
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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

−9.177560661036899782141696784791, −8.145727527971587122277823350413, −7.71082656898929472955197756735, −6.48824635068822236084354829909, −6.30813687282066532071911792318, −4.84997373225421163175518474488, −4.54567687485822805087611683761, −3.39450093463639182122679742165, −2.41422636937817928100301349482, −1.37640595930819042394200435957, 1.17176305778321780327489438332, 2.03039281902100178525800764842, 3.40199361499215932352142370020, 4.35242699206042715101628456548, 5.04061035419284437521122710031, 5.53565568904878562393455165706, 6.80230984952629457457825565117, 7.37343647748954598472541528072, 8.363444237725337152259799313833, 8.935606445969056321374218384710

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