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Properties

Label 2-114-57.50-c1-0-2
Degree $2$
Conductor $114$
Sign $0.211 - 0.977i$
Analytic cond. $0.910294$
Root an. cond. $0.954093$
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  + (0.5 + 0.866i)2-s + 1.73i·3-s + (−0.499 + 0.866i)4-s + (3 − 1.73i)5-s + (−1.49 + 0.866i)6-s − 2·7-s − 0.999·8-s − 2.99·9-s + (3 + 1.73i)10-s + 1.73i·11-s + (−1.49 − 0.866i)12-s + (−3 − 1.73i)13-s + (−1 − 1.73i)14-s + (2.99 + 5.19i)15-s + (−0.5 − 0.866i)16-s + (6 − 3.46i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + 0.999i·3-s + (−0.249 + 0.433i)4-s + (1.34 − 0.774i)5-s + (−0.612 + 0.353i)6-s − 0.755·7-s − 0.353·8-s − 0.999·9-s + (0.948 + 0.547i)10-s + 0.522i·11-s + (−0.433 − 0.249i)12-s + (−0.832 − 0.480i)13-s + (−0.267 − 0.462i)14-s + (0.774 + 1.34i)15-s + (−0.125 − 0.216i)16-s + (1.45 − 0.840i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.211 - 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.211 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(114\)    =    \(2 \cdot 3 \cdot 19\)
Sign: $0.211 - 0.977i$
Analytic conductor: \(0.910294\)
Root analytic conductor: \(0.954093\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{114} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 114,\ (\ :1/2),\ 0.211 - 0.977i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.00660 + 0.812287i\)
\(L(\frac12)\) \(\approx\) \(1.00660 + 0.812287i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 - 1.73iT \)
19 \( 1 + (-0.5 + 4.33i)T \)
good5 \( 1 + (-3 + 1.73i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + 2T + 7T^{2} \)
11 \( 1 - 1.73iT - 11T^{2} \)
13 \( 1 + (3 + 1.73i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-6 + 3.46i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (3 - 5.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 6.92iT - 31T^{2} \)
37 \( 1 - 6.92iT - 37T^{2} \)
41 \( 1 + (1.5 + 2.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4 + 6.92i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3 + 1.73i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3 + 5.19i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.5 - 2.59i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5 - 8.66i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.5 - 2.59i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3 - 5.19i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (2.5 + 4.33i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (12 - 6.92i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 5.19iT - 83T^{2} \)
89 \( 1 + (3 - 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-7.5 + 4.33i)T + (48.5 - 84.0i)T^{2} \)
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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

−13.92835200642126841447546786297, −12.98303247760488020421352443880, −12.02214196569113067546512546877, −10.15343836154507203378993261647, −9.649821626580960755746413617885, −8.708621184955583284032130744406, −6.97894429427254294578367153825, −5.47662583494971535382675742140, −4.98221653212859648678770452758, −3.05631423872479306912342985543, 1.95196628564662745654120187475, 3.24426703404144333301740499515, 5.76995324994470245826656273727, 6.28164355707810921589486350585, 7.75230985102298324916190910150, 9.491621912540238649998979429082, 10.16640429572740401937683306962, 11.44732127937116549735938712841, 12.55024267937959604068984170372, 13.26631228244715344175451384191

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