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Properties

Label 2-58-29.7-c1-0-1
Degree $2$
Conductor $58$
Sign $0.895 + 0.444i$
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  + (−0.222 − 0.974i)2-s + (1.56 + 0.755i)3-s + (−0.900 + 0.433i)4-s + (0.110 + 0.482i)5-s + (0.387 − 1.69i)6-s + (−2.82 − 1.36i)7-s + (0.623 + 0.781i)8-s + (0.0208 + 0.0261i)9-s + (0.445 − 0.214i)10-s + (0.870 − 1.09i)11-s − 1.74·12-s + (−3.56 + 4.46i)13-s + (−0.698 + 3.05i)14-s + (−0.191 + 0.840i)15-s + (0.623 − 0.781i)16-s + 5.31·17-s + ⋯
L(s)  = 1  + (−0.157 − 0.689i)2-s + (0.905 + 0.436i)3-s + (−0.450 + 0.216i)4-s + (0.0492 + 0.215i)5-s + (0.158 − 0.693i)6-s + (−1.06 − 0.514i)7-s + (0.220 + 0.276i)8-s + (0.00695 + 0.00871i)9-s + (0.140 − 0.0678i)10-s + (0.262 − 0.329i)11-s − 0.502·12-s + (−0.988 + 1.23i)13-s + (−0.186 + 0.817i)14-s + (−0.0495 + 0.216i)15-s + (0.155 − 0.195i)16-s + 1.28·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(58\)    =    \(2 \cdot 29\)
Sign: $0.895 + 0.444i$
Analytic conductor: \(0.463132\)
Root analytic conductor: \(0.680538\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{58} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 58,\ (\ :1/2),\ 0.895 + 0.444i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.888209 - 0.208146i\)
\(L(\frac12)\) \(\approx\) \(0.888209 - 0.208146i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.222 + 0.974i)T \)
29 \( 1 + (-5.38 - 0.0414i)T \)
good3 \( 1 + (-1.56 - 0.755i)T + (1.87 + 2.34i)T^{2} \)
5 \( 1 + (-0.110 - 0.482i)T + (-4.50 + 2.16i)T^{2} \)
7 \( 1 + (2.82 + 1.36i)T + (4.36 + 5.47i)T^{2} \)
11 \( 1 + (-0.870 + 1.09i)T + (-2.44 - 10.7i)T^{2} \)
13 \( 1 + (3.56 - 4.46i)T + (-2.89 - 12.6i)T^{2} \)
17 \( 1 - 5.31T + 17T^{2} \)
19 \( 1 + (4.47 - 2.15i)T + (11.8 - 14.8i)T^{2} \)
23 \( 1 + (-0.181 + 0.793i)T + (-20.7 - 9.97i)T^{2} \)
31 \( 1 + (1.41 + 6.20i)T + (-27.9 + 13.4i)T^{2} \)
37 \( 1 + (-5.56 - 6.97i)T + (-8.23 + 36.0i)T^{2} \)
41 \( 1 + 4.01T + 41T^{2} \)
43 \( 1 + (-0.310 + 1.36i)T + (-38.7 - 18.6i)T^{2} \)
47 \( 1 + (6.42 - 8.05i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (0.944 + 4.13i)T + (-47.7 + 22.9i)T^{2} \)
59 \( 1 - 11.1T + 59T^{2} \)
61 \( 1 + (4.38 + 2.11i)T + (38.0 + 47.6i)T^{2} \)
67 \( 1 + (-4.45 - 5.58i)T + (-14.9 + 65.3i)T^{2} \)
71 \( 1 + (-3.76 + 4.72i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (2.73 - 11.9i)T + (-65.7 - 31.6i)T^{2} \)
79 \( 1 + (5.86 + 7.35i)T + (-17.5 + 77.0i)T^{2} \)
83 \( 1 + (11.4 - 5.51i)T + (51.7 - 64.8i)T^{2} \)
89 \( 1 + (0.398 + 1.74i)T + (-80.1 + 38.6i)T^{2} \)
97 \( 1 + (3.42 - 1.64i)T + (60.4 - 75.8i)T^{2} \)
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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

−14.71496146257308266435565361037, −14.14561189836495382902830381805, −12.86781172681557728591155802493, −11.72650993109979044333554479108, −10.11306504593307121714581040135, −9.564520212765823918553394556079, −8.290700106888424980790488671685, −6.58664819165781313818070734103, −4.14835157466600069837477845827, −2.88304899228522888301033541176, 2.97289032003055873712764771288, 5.34179659423460159677096180889, 6.91261036887911701110291974698, 8.092593397426579649064035243793, 9.142357688520052268455155309236, 10.24135360093120599801891144439, 12.46742317437420017130985695710, 13.04591739612882840996438901497, 14.41780360727538971274792358777, 15.09727535894556795918395182625

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