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Properties

Label 2-58-29.25-c1-0-0
Degree $2$
Conductor $58$
Sign $-0.415 - 0.909i$
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 + (−2.06 + 0.996i)3-s + (−0.900 − 0.433i)4-s + (−0.788 + 3.45i)5-s + (−0.511 − 2.23i)6-s + (3.72 − 1.79i)7-s + (0.623 − 0.781i)8-s + (1.41 − 1.77i)9-s + (−3.19 − 1.53i)10-s + (−1.14 − 1.43i)11-s + 2.29·12-s + (2.09 + 2.62i)13-s + (0.920 + 4.03i)14-s + (−1.81 − 7.93i)15-s + (0.623 + 0.781i)16-s + 3.52·17-s + ⋯
L(s)  = 1  + (−0.157 + 0.689i)2-s + (−1.19 + 0.575i)3-s + (−0.450 − 0.216i)4-s + (−0.352 + 1.54i)5-s + (−0.208 − 0.914i)6-s + (1.40 − 0.678i)7-s + (0.220 − 0.276i)8-s + (0.472 − 0.592i)9-s + (−1.00 − 0.486i)10-s + (−0.346 − 0.434i)11-s + 0.662·12-s + (0.580 + 0.728i)13-s + (0.246 + 1.07i)14-s + (−0.467 − 2.04i)15-s + (0.155 + 0.195i)16-s + 0.853·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.415 - 0.909i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.415 - 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(58\)    =    \(2 \cdot 29\)
Sign: $-0.415 - 0.909i$
Analytic conductor: \(0.463132\)
Root analytic conductor: \(0.680538\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{58} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 58,\ (\ :1/2),\ -0.415 - 0.909i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.317559 + 0.494472i\)
\(L(\frac12)\) \(\approx\) \(0.317559 + 0.494472i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.127 + 5.38i)T \)
good3 \( 1 + (2.06 - 0.996i)T + (1.87 - 2.34i)T^{2} \)
5 \( 1 + (0.788 - 3.45i)T + (-4.50 - 2.16i)T^{2} \)
7 \( 1 + (-3.72 + 1.79i)T + (4.36 - 5.47i)T^{2} \)
11 \( 1 + (1.14 + 1.43i)T + (-2.44 + 10.7i)T^{2} \)
13 \( 1 + (-2.09 - 2.62i)T + (-2.89 + 12.6i)T^{2} \)
17 \( 1 - 3.52T + 17T^{2} \)
19 \( 1 + (2.45 + 1.18i)T + (11.8 + 14.8i)T^{2} \)
23 \( 1 + (-0.679 - 2.97i)T + (-20.7 + 9.97i)T^{2} \)
31 \( 1 + (0.196 - 0.861i)T + (-27.9 - 13.4i)T^{2} \)
37 \( 1 + (-3.04 + 3.82i)T + (-8.23 - 36.0i)T^{2} \)
41 \( 1 + 3.01T + 41T^{2} \)
43 \( 1 + (0.409 + 1.79i)T + (-38.7 + 18.6i)T^{2} \)
47 \( 1 + (-1.25 - 1.57i)T + (-10.4 + 45.8i)T^{2} \)
53 \( 1 + (-1.47 + 6.46i)T + (-47.7 - 22.9i)T^{2} \)
59 \( 1 - 6.12T + 59T^{2} \)
61 \( 1 + (1.64 - 0.792i)T + (38.0 - 47.6i)T^{2} \)
67 \( 1 + (0.0862 - 0.108i)T + (-14.9 - 65.3i)T^{2} \)
71 \( 1 + (8.17 + 10.2i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (-3.42 - 15.0i)T + (-65.7 + 31.6i)T^{2} \)
79 \( 1 + (9.90 - 12.4i)T + (-17.5 - 77.0i)T^{2} \)
83 \( 1 + (-0.0422 - 0.0203i)T + (51.7 + 64.8i)T^{2} \)
89 \( 1 + (-0.800 + 3.50i)T + (-80.1 - 38.6i)T^{2} \)
97 \( 1 + (4.71 + 2.27i)T + (60.4 + 75.8i)T^{2} \)
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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

−15.58908308883544098137437786873, −14.62280287628191765482244116299, −13.80880081826063560098985224234, −11.52127322663672973833712545796, −11.04558915990268185782307791155, −10.16137922679388978449127747726, −8.073009216166334142413631251446, −6.92607481815687046129847119977, −5.64285772551687805729814546390, −4.15671795440660988422127086633, 1.29498353440283692095818641178, 4.74589366603281866618749223747, 5.57893596623414583010741519341, 7.913740907565583237728391946368, 8.772224260701244701638968545128, 10.60947695646583784624948794785, 11.72776195617045305993113197747, 12.31806778384103090564873576078, 13.07086879910951440272918348277, 14.86146338082526352504848458476

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