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Properties

Label 2-58-29.25-c1-0-2
Degree $2$
Conductor $58$
Sign $0.655 + 0.755i$
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 + (0.5 − 0.240i)3-s + (−0.900 − 0.433i)4-s + (−0.0440 + 0.193i)5-s + (−0.123 − 0.541i)6-s + (−0.0990 + 0.0476i)7-s + (−0.623 + 0.781i)8-s + (−1.67 + 2.10i)9-s + (0.178 + 0.0859i)10-s + (−0.832 − 1.04i)11-s − 0.554·12-s + (2.45 + 3.07i)13-s + (0.0244 + 0.107i)14-s + (0.0244 + 0.107i)15-s + (0.623 + 0.781i)16-s − 2.91·17-s + ⋯
L(s)  = 1  + (0.157 − 0.689i)2-s + (0.288 − 0.139i)3-s + (−0.450 − 0.216i)4-s + (−0.0197 + 0.0863i)5-s + (−0.0504 − 0.220i)6-s + (−0.0374 + 0.0180i)7-s + (−0.220 + 0.276i)8-s + (−0.559 + 0.701i)9-s + (0.0564 + 0.0271i)10-s + (−0.250 − 0.314i)11-s − 0.160·12-s + (0.681 + 0.854i)13-s + (0.00653 + 0.0286i)14-s + (0.00631 + 0.0276i)15-s + (0.155 + 0.195i)16-s − 0.706·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.655 + 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.655 + 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(58\)    =    \(2 \cdot 29\)
Sign: $0.655 + 0.755i$
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: \((2,\ 58,\ (\ :1/2),\ 0.655 + 0.755i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.847978 - 0.386899i\)
\(L(\frac12)\) \(\approx\) \(0.847978 - 0.386899i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.39 + 5.20i)T \)
good3 \( 1 + (-0.5 + 0.240i)T + (1.87 - 2.34i)T^{2} \)
5 \( 1 + (0.0440 - 0.193i)T + (-4.50 - 2.16i)T^{2} \)
7 \( 1 + (0.0990 - 0.0476i)T + (4.36 - 5.47i)T^{2} \)
11 \( 1 + (0.832 + 1.04i)T + (-2.44 + 10.7i)T^{2} \)
13 \( 1 + (-2.45 - 3.07i)T + (-2.89 + 12.6i)T^{2} \)
17 \( 1 + 2.91T + 17T^{2} \)
19 \( 1 + (1.16 + 0.562i)T + (11.8 + 14.8i)T^{2} \)
23 \( 1 + (1.73 + 7.59i)T + (-20.7 + 9.97i)T^{2} \)
31 \( 1 + (-2.07 + 9.11i)T + (-27.9 - 13.4i)T^{2} \)
37 \( 1 + (1.88 - 2.36i)T + (-8.23 - 36.0i)T^{2} \)
41 \( 1 - 3.76T + 41T^{2} \)
43 \( 1 + (-1.48 - 6.49i)T + (-38.7 + 18.6i)T^{2} \)
47 \( 1 + (-0.5 - 0.626i)T + (-10.4 + 45.8i)T^{2} \)
53 \( 1 + (1.85 - 8.12i)T + (-47.7 - 22.9i)T^{2} \)
59 \( 1 + 5.08T + 59T^{2} \)
61 \( 1 + (-9.96 + 4.79i)T + (38.0 - 47.6i)T^{2} \)
67 \( 1 + (6.85 - 8.60i)T + (-14.9 - 65.3i)T^{2} \)
71 \( 1 + (-6.82 - 8.56i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (1.76 + 7.74i)T + (-65.7 + 31.6i)T^{2} \)
79 \( 1 + (-3.04 + 3.82i)T + (-17.5 - 77.0i)T^{2} \)
83 \( 1 + (-10.5 - 5.06i)T + (51.7 + 64.8i)T^{2} \)
89 \( 1 + (-2.55 + 11.2i)T + (-80.1 - 38.6i)T^{2} \)
97 \( 1 + (-9.54 - 4.59i)T + (60.4 + 75.8i)T^{2} \)
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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

−14.73955296229157632588062285675, −13.76115648170781114840128787761, −12.94081722769592167432755401907, −11.47201914863645294313998916891, −10.72108667218927601689804945080, −9.174178360869419454189491062449, −8.107000450416913811443947421351, −6.22411541886194848638701809957, −4.42542696393205624443681899490, −2.51267149914475230714086821315, 3.49357046997190486475942377954, 5.32943593020712940336867539766, 6.72739288050964183111280694597, 8.225239511846850158540244353264, 9.191521501729499755706783795968, 10.67215851392972238258172563528, 12.20694610435273642423391662469, 13.30837238118236334187272276555, 14.40694218610462715111579306294, 15.36146466461543118137215038524

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