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Properties

Label 2-580-580.3-c0-0-0
Degree $2$
Conductor $580$
Sign $0.973 + 0.227i$
Analytic cond. $0.289457$
Root an. cond. $0.538012$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.974 + 0.222i)2-s + (0.900 − 0.433i)4-s + (0.433 − 0.900i)5-s + (−0.781 + 0.623i)8-s + (0.623 + 0.781i)9-s + (−0.222 + 0.974i)10-s + (0.0739 + 0.656i)13-s + (0.623 − 0.781i)16-s − 1.94i·17-s + (−0.781 − 0.623i)18-s − i·20-s + (−0.623 − 0.781i)25-s + (−0.218 − 0.623i)26-s + (0.781 + 0.623i)29-s + (−0.433 + 0.900i)32-s + ⋯
L(s)  = 1  + (−0.974 + 0.222i)2-s + (0.900 − 0.433i)4-s + (0.433 − 0.900i)5-s + (−0.781 + 0.623i)8-s + (0.623 + 0.781i)9-s + (−0.222 + 0.974i)10-s + (0.0739 + 0.656i)13-s + (0.623 − 0.781i)16-s − 1.94i·17-s + (−0.781 − 0.623i)18-s − i·20-s + (−0.623 − 0.781i)25-s + (−0.218 − 0.623i)26-s + (0.781 + 0.623i)29-s + (−0.433 + 0.900i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 580 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.227i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 580 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.227i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(580\)    =    \(2^{2} \cdot 5 \cdot 29\)
Sign: $0.973 + 0.227i$
Analytic conductor: \(0.289457\)
Root analytic conductor: \(0.538012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{580} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 580,\ (\ :0),\ 0.973 + 0.227i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6758671724\)
\(L(\frac12)\) \(\approx\) \(0.6758671724\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.974 - 0.222i)T \)
5 \( 1 + (-0.433 + 0.900i)T \)
29 \( 1 + (-0.781 - 0.623i)T \)
good3 \( 1 + (-0.623 - 0.781i)T^{2} \)
7 \( 1 + (-0.781 + 0.623i)T^{2} \)
11 \( 1 + (0.974 - 0.222i)T^{2} \)
13 \( 1 + (-0.0739 - 0.656i)T + (-0.974 + 0.222i)T^{2} \)
17 \( 1 + 1.94iT - T^{2} \)
19 \( 1 + (-0.781 - 0.623i)T^{2} \)
23 \( 1 + (-0.433 + 0.900i)T^{2} \)
31 \( 1 + (-0.433 - 0.900i)T^{2} \)
37 \( 1 + (-1.12 - 1.40i)T + (-0.222 + 0.974i)T^{2} \)
41 \( 1 + (1.33 + 1.33i)T + iT^{2} \)
43 \( 1 + (0.900 + 0.433i)T^{2} \)
47 \( 1 + (0.222 + 0.974i)T^{2} \)
53 \( 1 + (0.566 - 0.900i)T + (-0.433 - 0.900i)T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + (1.59 - 0.559i)T + (0.781 - 0.623i)T^{2} \)
67 \( 1 + (0.974 + 0.222i)T^{2} \)
71 \( 1 + (-0.222 - 0.974i)T^{2} \)
73 \( 1 + (0.433 + 0.0990i)T + (0.900 + 0.433i)T^{2} \)
79 \( 1 + (0.974 + 0.222i)T^{2} \)
83 \( 1 + (-0.781 - 0.623i)T^{2} \)
89 \( 1 + (1.05 - 1.68i)T + (-0.433 - 0.900i)T^{2} \)
97 \( 1 + (0.781 - 0.376i)T + (0.623 - 0.781i)T^{2} \)
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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

−10.64593130410493557221926003843, −9.818157109211567959923238594485, −9.191259372817523253695266683254, −8.380114174242819171004951993073, −7.42274457321616635425387424862, −6.63789983539076172326376544139, −5.37406339500333712396953801046, −4.61390382933098271482369573623, −2.61897881850716786709465327375, −1.35833975393823302682912603802, 1.61442408924215957215912133969, 2.96653439068943675524219499991, 3.97785223895348764154886052324, 5.99354848338282178708247553256, 6.46903040996226286106166433456, 7.52671728901673121945235712072, 8.330783441536245466293252554503, 9.402563101604943090062803475103, 10.15423809263998371207669537349, 10.64835153304434470130885556805

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