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Properties

Label 2-58-29.23-c1-0-1
Degree $2$
Conductor $58$
Sign $0.939 + 0.342i$
Analytic cond. $0.463132$
Root an. cond. $0.680538$
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  + (−0.623 + 0.781i)2-s + (0.5 − 2.19i)3-s + (−0.222 − 0.974i)4-s + (0.969 − 1.21i)5-s + (1.40 + 1.75i)6-s + (−0.777 + 3.40i)7-s + (0.900 + 0.433i)8-s + (−1.84 − 0.888i)9-s + (0.346 + 1.51i)10-s + (−3.37 + 1.62i)11-s − 2.24·12-s + (3.46 − 1.67i)13-s + (−2.17 − 2.73i)14-s + (−2.17 − 2.73i)15-s + (−0.900 + 0.433i)16-s − 4.93·17-s + ⋯
L(s)  = 1  + (−0.440 + 0.552i)2-s + (0.288 − 1.26i)3-s + (−0.111 − 0.487i)4-s + (0.433 − 0.543i)5-s + (0.571 + 0.717i)6-s + (−0.293 + 1.28i)7-s + (0.318 + 0.153i)8-s + (−0.615 − 0.296i)9-s + (0.109 + 0.479i)10-s + (−1.01 + 0.489i)11-s − 0.648·12-s + (0.962 − 0.463i)13-s + (−0.582 − 0.730i)14-s + (−0.562 − 0.705i)15-s + (−0.225 + 0.108i)16-s − 1.19·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(58\)    =    \(2 \cdot 29\)
Sign: $0.939 + 0.342i$
Analytic conductor: \(0.463132\)
Root analytic conductor: \(0.680538\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{58} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 58,\ (\ :1/2),\ 0.939 + 0.342i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.764208 - 0.134871i\)
\(L(\frac12)\) \(\approx\) \(0.764208 - 0.134871i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.623 - 0.781i)T \)
29 \( 1 + (-5.38 - 0.202i)T \)
good3 \( 1 + (-0.5 + 2.19i)T + (-2.70 - 1.30i)T^{2} \)
5 \( 1 + (-0.969 + 1.21i)T + (-1.11 - 4.87i)T^{2} \)
7 \( 1 + (0.777 - 3.40i)T + (-6.30 - 3.03i)T^{2} \)
11 \( 1 + (3.37 - 1.62i)T + (6.85 - 8.60i)T^{2} \)
13 \( 1 + (-3.46 + 1.67i)T + (8.10 - 10.1i)T^{2} \)
17 \( 1 + 4.93T + 17T^{2} \)
19 \( 1 + (-1.37 - 6.00i)T + (-17.1 + 8.24i)T^{2} \)
23 \( 1 + (3.59 + 4.50i)T + (-5.11 + 22.4i)T^{2} \)
31 \( 1 + (-1.56 + 1.96i)T + (-6.89 - 30.2i)T^{2} \)
37 \( 1 + (6.11 + 2.94i)T + (23.0 + 28.9i)T^{2} \)
41 \( 1 - 0.0271T + 41T^{2} \)
43 \( 1 + (7.32 + 9.17i)T + (-9.56 + 41.9i)T^{2} \)
47 \( 1 + (-0.5 + 0.240i)T + (29.3 - 36.7i)T^{2} \)
53 \( 1 + (-2.03 + 2.54i)T + (-11.7 - 51.6i)T^{2} \)
59 \( 1 + 3.06T + 59T^{2} \)
61 \( 1 + (0.0293 - 0.128i)T + (-54.9 - 26.4i)T^{2} \)
67 \( 1 + (-9.91 - 4.77i)T + (41.7 + 52.3i)T^{2} \)
71 \( 1 + (-3.26 + 1.57i)T + (44.2 - 55.5i)T^{2} \)
73 \( 1 + (4.13 + 5.18i)T + (-16.2 + 71.1i)T^{2} \)
79 \( 1 + (1.35 + 0.653i)T + (49.2 + 61.7i)T^{2} \)
83 \( 1 + (-2.21 - 9.71i)T + (-74.7 + 36.0i)T^{2} \)
89 \( 1 + (3.36 - 4.21i)T + (-19.8 - 86.7i)T^{2} \)
97 \( 1 + (0.960 + 4.20i)T + (-87.3 + 42.0i)T^{2} \)
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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

−15.33716332037684184225007235081, −13.84519553070315860604020607899, −12.95234467415586498595036100171, −12.16808690220093007946945936651, −10.27538024602884455279864693608, −8.773093694927719344852321470889, −8.086854333327744959624170888822, −6.56402969572114070863579091511, −5.47263143702377533155607807990, −2.10258903573578906270884423837, 3.16637332030411344290933635322, 4.51837452040187098641598730832, 6.77376692475317745727957685680, 8.510873813681926101157526716007, 9.760276743025015018212815437882, 10.53964693326956809129642871226, 11.20565428856014023386998909260, 13.37931210555110664205752241695, 13.88937702853558254055406337791, 15.62254209896177964077465366182

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