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Properties

Label 2-58-29.4-c1-0-0
Degree $2$
Conductor $58$
Sign $0.863 - 0.505i$
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  + (−0.974 − 0.222i)2-s + (0.626 + 1.30i)3-s + (0.900 + 0.433i)4-s + (−0.308 + 1.35i)5-s + (−0.321 − 1.40i)6-s + (1.78 − 0.861i)7-s + (−0.781 − 0.623i)8-s + (0.568 − 0.712i)9-s + (0.601 − 1.24i)10-s + (−2.70 + 2.15i)11-s + 1.44i·12-s + (−3.81 − 4.77i)13-s + (−1.93 + 0.442i)14-s + (−1.95 + 0.445i)15-s + (0.623 + 0.781i)16-s − 4.25i·17-s + ⋯
L(s)  = 1  + (−0.689 − 0.157i)2-s + (0.361 + 0.751i)3-s + (0.450 + 0.216i)4-s + (−0.137 + 0.604i)5-s + (−0.131 − 0.575i)6-s + (0.676 − 0.325i)7-s + (−0.276 − 0.220i)8-s + (0.189 − 0.237i)9-s + (0.190 − 0.394i)10-s + (−0.814 + 0.649i)11-s + 0.417i·12-s + (−1.05 − 1.32i)13-s + (−0.517 + 0.118i)14-s + (−0.504 + 0.115i)15-s + (0.155 + 0.195i)16-s − 1.03i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.863 - 0.505i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.863 - 0.505i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(58\)    =    \(2 \cdot 29\)
Sign: $0.863 - 0.505i$
Analytic conductor: \(0.463132\)
Root analytic conductor: \(0.680538\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{58} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 58,\ (\ :1/2),\ 0.863 - 0.505i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.699729 + 0.189730i\)
\(L(\frac12)\) \(\approx\) \(0.699729 + 0.189730i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.974 + 0.222i)T \)
29 \( 1 + (4.17 + 3.40i)T \)
good3 \( 1 + (-0.626 - 1.30i)T + (-1.87 + 2.34i)T^{2} \)
5 \( 1 + (0.308 - 1.35i)T + (-4.50 - 2.16i)T^{2} \)
7 \( 1 + (-1.78 + 0.861i)T + (4.36 - 5.47i)T^{2} \)
11 \( 1 + (2.70 - 2.15i)T + (2.44 - 10.7i)T^{2} \)
13 \( 1 + (3.81 + 4.77i)T + (-2.89 + 12.6i)T^{2} \)
17 \( 1 + 4.25iT - 17T^{2} \)
19 \( 1 + (0.940 - 1.95i)T + (-11.8 - 14.8i)T^{2} \)
23 \( 1 + (-0.127 - 0.556i)T + (-20.7 + 9.97i)T^{2} \)
31 \( 1 + (-5.88 - 1.34i)T + (27.9 + 13.4i)T^{2} \)
37 \( 1 + (2.12 + 1.69i)T + (8.23 + 36.0i)T^{2} \)
41 \( 1 - 9.14iT - 41T^{2} \)
43 \( 1 + (10.7 - 2.46i)T + (38.7 - 18.6i)T^{2} \)
47 \( 1 + (3.89 - 3.10i)T + (10.4 - 45.8i)T^{2} \)
53 \( 1 + (-2.32 + 10.1i)T + (-47.7 - 22.9i)T^{2} \)
59 \( 1 - 9.01T + 59T^{2} \)
61 \( 1 + (-4.16 - 8.64i)T + (-38.0 + 47.6i)T^{2} \)
67 \( 1 + (7.82 - 9.81i)T + (-14.9 - 65.3i)T^{2} \)
71 \( 1 + (2.66 + 3.34i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (-6.80 + 1.55i)T + (65.7 - 31.6i)T^{2} \)
79 \( 1 + (9.09 + 7.25i)T + (17.5 + 77.0i)T^{2} \)
83 \( 1 + (-9.76 - 4.70i)T + (51.7 + 64.8i)T^{2} \)
89 \( 1 + (2.78 + 0.634i)T + (80.1 + 38.6i)T^{2} \)
97 \( 1 + (1.05 - 2.18i)T + (-60.4 - 75.8i)T^{2} \)
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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

−15.13728038663277387593742098598, −14.73321124051844137234097681855, −12.96523139826998736331195466402, −11.58593935244096194481951383019, −10.29737400718234137986458624291, −9.816370901908045352300578843734, −8.141033604085866630855223395902, −7.15508496456974016707390823056, −4.88175578109012888090872672260, −2.95412219451958159327794785061, 2.02862561492139606548171251717, 4.97360123664242439891011969067, 6.85915834948548848428431638868, 8.081200169120690533548154079707, 8.815452540761104622005400492832, 10.41453237763452535379438099440, 11.74596238501443488128019185687, 12.82052213848433658319168717174, 14.00003912717072950025257867463, 15.15767747313980539985836955715

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