Properties

Label 2-114-57.2-c1-0-0
Degree $2$
Conductor $114$
Sign $-0.509 - 0.860i$
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  + (−0.173 + 0.984i)2-s + (−0.0553 + 1.73i)3-s + (−0.939 − 0.342i)4-s + (0.882 + 2.42i)5-s + (−1.69 − 0.355i)6-s + (−1.58 − 2.74i)7-s + (0.5 − 0.866i)8-s + (−2.99 − 0.191i)9-s + (−2.54 + 0.448i)10-s + (2.16 + 1.25i)11-s + (0.644 − 1.60i)12-s + (2.71 + 3.24i)13-s + (2.97 − 1.08i)14-s + (−4.24 + 1.39i)15-s + (0.766 + 0.642i)16-s + (−1.32 − 0.233i)17-s + ⋯
L(s)  = 1  + (−0.122 + 0.696i)2-s + (−0.0319 + 0.999i)3-s + (−0.469 − 0.171i)4-s + (0.394 + 1.08i)5-s + (−0.692 − 0.144i)6-s + (−0.598 − 1.03i)7-s + (0.176 − 0.306i)8-s + (−0.997 − 0.0638i)9-s + (−0.803 + 0.141i)10-s + (0.653 + 0.377i)11-s + (0.185 − 0.464i)12-s + (0.754 + 0.898i)13-s + (0.795 − 0.289i)14-s + (−1.09 + 0.359i)15-s + (0.191 + 0.160i)16-s + (−0.320 − 0.0565i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.509 - 0.860i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.509 - 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(114\)    =    \(2 \cdot 3 \cdot 19\)
Sign: $-0.509 - 0.860i$
Analytic conductor: \(0.910294\)
Root analytic conductor: \(0.954093\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{114} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 114,\ (\ :1/2),\ -0.509 - 0.860i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.467984 + 0.820436i\)
\(L(\frac12)\) \(\approx\) \(0.467984 + 0.820436i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.173 - 0.984i)T \)
3 \( 1 + (0.0553 - 1.73i)T \)
19 \( 1 + (-3.14 + 3.01i)T \)
good5 \( 1 + (-0.882 - 2.42i)T + (-3.83 + 3.21i)T^{2} \)
7 \( 1 + (1.58 + 2.74i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.16 - 1.25i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.71 - 3.24i)T + (-2.25 + 12.8i)T^{2} \)
17 \( 1 + (1.32 + 0.233i)T + (15.9 + 5.81i)T^{2} \)
23 \( 1 + (-1.30 + 3.58i)T + (-17.6 - 14.7i)T^{2} \)
29 \( 1 + (-1.32 - 7.49i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (-6.89 + 3.97i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 4.10iT - 37T^{2} \)
41 \( 1 + (4.95 + 4.16i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (11.7 - 4.27i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (6.16 - 1.08i)T + (44.1 - 16.0i)T^{2} \)
53 \( 1 + (-3.46 - 1.26i)T + (40.6 + 34.0i)T^{2} \)
59 \( 1 + (-1.54 + 8.75i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (0.133 + 0.0485i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (4.48 - 0.791i)T + (62.9 - 22.9i)T^{2} \)
71 \( 1 + (8.59 - 3.12i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 + (1.67 + 1.40i)T + (12.6 + 71.8i)T^{2} \)
79 \( 1 + (-6.41 + 7.64i)T + (-13.7 - 77.7i)T^{2} \)
83 \( 1 + (12.3 - 7.11i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-12.7 + 10.6i)T + (15.4 - 87.6i)T^{2} \)
97 \( 1 + (-0.538 - 0.0949i)T + (91.1 + 33.1i)T^{2} \)
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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

−14.16726128166956545024856805643, −13.42469319451217603352215894409, −11.52100418535152963633926255671, −10.52116882501486052384002199281, −9.778907939202467104685595414537, −8.755243760763336598851563294190, −6.96137578821887347878257558851, −6.40285839495024863312887549580, −4.62493903445833057641154411904, −3.37288811449141544272358512240, 1.33964862873533353556297718732, 3.12506621034275983850418507779, 5.33037507592124389968303135061, 6.29406986463138430768377161935, 8.220852541580195392015481680472, 8.834768417426439600105273640380, 9.951551411675276440184342894607, 11.67241860314474014857482985384, 12.12332704479552032515041531310, 13.25344761853728693417516119854

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