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Properties

Label 2-114-57.56-c1-0-2
Degree $2$
Conductor $114$
Sign $0.993 - 0.114i$
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  − 2-s + (1.5 − 0.866i)3-s + 4-s + 3.46i·5-s + (−1.5 + 0.866i)6-s + 7-s − 8-s + (1.5 − 2.59i)9-s − 3.46i·10-s − 3.46i·11-s + (1.5 − 0.866i)12-s + 1.73i·13-s − 14-s + (2.99 + 5.19i)15-s + 16-s + 1.73i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.866 − 0.499i)3-s + 0.5·4-s + 1.54i·5-s + (−0.612 + 0.353i)6-s + 0.377·7-s − 0.353·8-s + (0.5 − 0.866i)9-s − 1.09i·10-s − 1.04i·11-s + (0.433 − 0.249i)12-s + 0.480i·13-s − 0.267·14-s + (0.774 + 1.34i)15-s + 0.250·16-s + 0.420i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.114i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 - 0.114i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(114\)    =    \(2 \cdot 3 \cdot 19\)
Sign: $0.993 - 0.114i$
Analytic conductor: \(0.910294\)
Root analytic conductor: \(0.954093\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{114} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 114,\ (\ :1/2),\ 0.993 - 0.114i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.00574 + 0.0578745i\)
\(L(\frac12)\) \(\approx\) \(1.00574 + 0.0578745i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.5 + 0.866i)T \)
19 \( 1 + (4 + 1.73i)T \)
good5 \( 1 - 3.46iT - 5T^{2} \)
7 \( 1 - T + 7T^{2} \)
11 \( 1 + 3.46iT - 11T^{2} \)
13 \( 1 - 1.73iT - 13T^{2} \)
17 \( 1 - 1.73iT - 17T^{2} \)
23 \( 1 + 5.19iT - 23T^{2} \)
29 \( 1 + 9T + 29T^{2} \)
31 \( 1 - 10.3iT - 31T^{2} \)
37 \( 1 + 6.92iT - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 2T + 43T^{2} \)
47 \( 1 - 3.46iT - 47T^{2} \)
53 \( 1 - 9T + 53T^{2} \)
59 \( 1 - 3T + 59T^{2} \)
61 \( 1 + 8T + 61T^{2} \)
67 \( 1 + 8.66iT - 67T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 - 11T + 73T^{2} \)
79 \( 1 + 6.92iT - 79T^{2} \)
83 \( 1 - 10.3iT - 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 - 13.8iT - 97T^{2} \)
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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

−13.94405735128223976422455890320, −12.57579759618386388604901078259, −11.15459925579274283049481845686, −10.58443456365459907580287938768, −9.155950241897008087526607507583, −8.212552959238264029829531990035, −7.10187701398086923190915939762, −6.29901505145278870809570277864, −3.54725453331496676552181256947, −2.24970756960208927516047071219, 1.89648213860845155691770044950, 4.13413131291149390841434116951, 5.35039909813264016226809681016, 7.54327311687729818305855281059, 8.307453311780821382709945834705, 9.314696248225270502049248431688, 9.930035272462788786741653902004, 11.39982173113675917207642660543, 12.66236941589837179367898930518, 13.41443002050458089445793186250

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