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Properties

Label 2-58-29.13-c1-0-0
Degree $2$
Conductor $58$
Sign $0.471 - 0.881i$
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  + (−0.433 + 0.900i)2-s + (2.19 + 1.74i)3-s + (−0.623 − 0.781i)4-s + (−3.43 − 1.65i)5-s + (−2.52 + 1.21i)6-s + (1.36 − 1.71i)7-s + (0.974 − 0.222i)8-s + (1.07 + 4.72i)9-s + (2.98 − 2.38i)10-s + (−0.0647 − 0.0147i)11-s − 2.80i·12-s + (−0.157 + 0.687i)13-s + (0.949 + 1.97i)14-s + (−4.64 − 9.63i)15-s + (−0.222 + 0.974i)16-s − 3.46i·17-s + ⋯
L(s)  = 1  + (−0.306 + 0.637i)2-s + (1.26 + 1.00i)3-s + (−0.311 − 0.390i)4-s + (−1.53 − 0.740i)5-s + (−1.03 + 0.496i)6-s + (0.515 − 0.646i)7-s + (0.344 − 0.0786i)8-s + (0.359 + 1.57i)9-s + (0.943 − 0.752i)10-s + (−0.0195 − 0.00445i)11-s − 0.808i·12-s + (−0.0435 + 0.190i)13-s + (0.253 + 0.526i)14-s + (−1.19 − 2.48i)15-s + (−0.0556 + 0.243i)16-s − 0.839i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.471 - 0.881i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.471 - 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(58\)    =    \(2 \cdot 29\)
Sign: $0.471 - 0.881i$
Analytic conductor: \(0.463132\)
Root analytic conductor: \(0.680538\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{58} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 58,\ (\ :1/2),\ 0.471 - 0.881i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.755380 + 0.452608i\)
\(L(\frac12)\) \(\approx\) \(0.755380 + 0.452608i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.433 - 0.900i)T \)
29 \( 1 + (-2.16 - 4.92i)T \)
good3 \( 1 + (-2.19 - 1.74i)T + (0.667 + 2.92i)T^{2} \)
5 \( 1 + (3.43 + 1.65i)T + (3.11 + 3.90i)T^{2} \)
7 \( 1 + (-1.36 + 1.71i)T + (-1.55 - 6.82i)T^{2} \)
11 \( 1 + (0.0647 + 0.0147i)T + (9.91 + 4.77i)T^{2} \)
13 \( 1 + (0.157 - 0.687i)T + (-11.7 - 5.64i)T^{2} \)
17 \( 1 + 3.46iT - 17T^{2} \)
19 \( 1 + (2.15 - 1.72i)T + (4.22 - 18.5i)T^{2} \)
23 \( 1 + (5.68 - 2.73i)T + (14.3 - 17.9i)T^{2} \)
31 \( 1 + (-3.24 + 6.74i)T + (-19.3 - 24.2i)T^{2} \)
37 \( 1 + (8.31 - 1.89i)T + (33.3 - 16.0i)T^{2} \)
41 \( 1 - 2.48iT - 41T^{2} \)
43 \( 1 + (0.624 + 1.29i)T + (-26.8 + 33.6i)T^{2} \)
47 \( 1 + (-8.77 - 2.00i)T + (42.3 + 20.3i)T^{2} \)
53 \( 1 + (-2.90 - 1.40i)T + (33.0 + 41.4i)T^{2} \)
59 \( 1 - 1.24T + 59T^{2} \)
61 \( 1 + (-2.61 - 2.08i)T + (13.5 + 59.4i)T^{2} \)
67 \( 1 + (1.56 + 6.83i)T + (-60.3 + 29.0i)T^{2} \)
71 \( 1 + (-2.24 + 9.84i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (-1.31 - 2.73i)T + (-45.5 + 57.0i)T^{2} \)
79 \( 1 + (-2.17 + 0.497i)T + (71.1 - 34.2i)T^{2} \)
83 \( 1 + (3.82 + 4.79i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (-7.59 + 15.7i)T + (-55.4 - 69.5i)T^{2} \)
97 \( 1 + (1.72 - 1.37i)T + (21.5 - 94.5i)T^{2} \)
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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

−15.56627221977377300087634536906, −14.56332265420272088652104753752, −13.66238402314968807641065588515, −11.92368776132151169313305339790, −10.48689436724372215945374906664, −9.174340012208870987245927808779, −8.230017133506104101219716494483, −7.52660254300522152994342854592, −4.71804452420580215597278671236, −3.83227926987666197740733034857, 2.43973740234156098959406793371, 3.85089157060427079795415542081, 6.95558415390046699058583900476, 8.156113741894002568085331732779, 8.550469245682265001244106076945, 10.52304575358119410725165112695, 11.86679795551209202203146726032, 12.49037408651506710036756494636, 13.93121470601692306879404371545, 14.86224018941700166606202162060

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