Properties

Label 2-114-57.41-c1-0-2
Degree $2$
Conductor $114$
Sign $0.999 + 0.0241i$
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  + (−0.766 − 0.642i)2-s + (1.57 + 0.715i)3-s + (0.173 + 0.984i)4-s + (0.262 + 0.0462i)5-s + (−0.748 − 1.56i)6-s + (0.604 + 1.04i)7-s + (0.500 − 0.866i)8-s + (1.97 + 2.25i)9-s + (−0.171 − 0.204i)10-s + (−2.03 − 1.17i)11-s + (−0.431 + 1.67i)12-s + (1.01 − 2.79i)13-s + (0.209 − 1.19i)14-s + (0.380 + 0.260i)15-s + (−0.939 + 0.342i)16-s + (−0.576 + 0.687i)17-s + ⋯
L(s)  = 1  + (−0.541 − 0.454i)2-s + (0.910 + 0.413i)3-s + (0.0868 + 0.492i)4-s + (0.117 + 0.0206i)5-s + (−0.305 − 0.637i)6-s + (0.228 + 0.395i)7-s + (0.176 − 0.306i)8-s + (0.658 + 0.752i)9-s + (−0.0541 − 0.0645i)10-s + (−0.613 − 0.354i)11-s + (−0.124 + 0.484i)12-s + (0.282 − 0.775i)13-s + (0.0561 − 0.318i)14-s + (0.0982 + 0.0673i)15-s + (−0.234 + 0.0855i)16-s + (−0.139 + 0.166i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0241i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0241i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(114\)    =    \(2 \cdot 3 \cdot 19\)
Sign: $0.999 + 0.0241i$
Analytic conductor: \(0.910294\)
Root analytic conductor: \(0.954093\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{114} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 114,\ (\ :1/2),\ 0.999 + 0.0241i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06692 - 0.0128935i\)
\(L(\frac12)\) \(\approx\) \(1.06692 - 0.0128935i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.766 + 0.642i)T \)
3 \( 1 + (-1.57 - 0.715i)T \)
19 \( 1 + (-1.97 + 3.88i)T \)
good5 \( 1 + (-0.262 - 0.0462i)T + (4.69 + 1.71i)T^{2} \)
7 \( 1 + (-0.604 - 1.04i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.03 + 1.17i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.01 + 2.79i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 + (0.576 - 0.687i)T + (-2.95 - 16.7i)T^{2} \)
23 \( 1 + (5.53 - 0.976i)T + (21.6 - 7.86i)T^{2} \)
29 \( 1 + (1.92 - 1.61i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (8.98 - 5.18i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 3.95iT - 37T^{2} \)
41 \( 1 + (-10.4 + 3.79i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (0.834 - 4.73i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (1.24 + 1.48i)T + (-8.16 + 46.2i)T^{2} \)
53 \( 1 + (0.998 + 5.66i)T + (-49.8 + 18.1i)T^{2} \)
59 \( 1 + (-9.78 - 8.20i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (0.153 + 0.871i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (-3.28 - 3.91i)T + (-11.6 + 65.9i)T^{2} \)
71 \( 1 + (-1.64 + 9.33i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (-0.320 + 0.116i)T + (55.9 - 46.9i)T^{2} \)
79 \( 1 + (-2.33 - 6.42i)T + (-60.5 + 50.7i)T^{2} \)
83 \( 1 + (12.2 - 7.05i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-11.5 - 4.20i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (-3.88 + 4.62i)T + (-16.8 - 95.5i)T^{2} \)
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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

−13.51834706136915056364000517282, −12.65097043707213186479914021116, −11.22723815833138819813335882005, −10.33478697596032712243500157653, −9.301513464787916902509922604763, −8.379799944519660073195976736612, −7.46294068282164518343701569017, −5.45156786676626132512074077102, −3.69866856239145669790281304119, −2.30814860113295529854540136606, 1.95500344041229009937522224334, 4.04347636289365385774522841437, 5.95252199854170901972792025127, 7.35808614611927033180319035084, 7.994121717789706914512428924240, 9.258300367719379830078048689807, 10.05346954081318253380467791551, 11.46018944105724170961199864962, 12.79668619264856808143978261602, 13.84422615256430038716175106879

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