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Properties

Label 2-58-29.7-c1-0-2
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: $\chi_{58} (7, \cdot )$
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

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

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