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Properties

Label 2-555-37.33-c1-0-1
Degree $2$
Conductor $555$
Sign $0.242 - 0.970i$
Analytic cond. $4.43169$
Root an. cond. $2.10515$
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  + (0.374 − 2.12i)2-s + (0.173 + 0.984i)3-s + (−2.50 − 0.910i)4-s + (−0.766 + 0.642i)5-s + 2.15·6-s + (−3.21 + 2.69i)7-s + (−0.713 + 1.23i)8-s + (−0.939 + 0.342i)9-s + (1.07 + 1.86i)10-s + (0.363 − 0.629i)11-s + (0.462 − 2.62i)12-s + (−4.15 − 1.51i)13-s + (4.52 + 7.84i)14-s + (−0.766 − 0.642i)15-s + (−1.71 − 1.43i)16-s + (−5.79 + 2.10i)17-s + ⋯
L(s)  = 1  + (0.265 − 1.50i)2-s + (0.100 + 0.568i)3-s + (−1.25 − 0.455i)4-s + (−0.342 + 0.287i)5-s + 0.881·6-s + (−1.21 + 1.01i)7-s + (−0.252 + 0.437i)8-s + (−0.313 + 0.114i)9-s + (0.341 + 0.591i)10-s + (0.109 − 0.189i)11-s + (0.133 − 0.756i)12-s + (−1.15 − 0.418i)13-s + (1.21 + 2.09i)14-s + (−0.197 − 0.165i)15-s + (−0.428 − 0.359i)16-s + (−1.40 + 0.511i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 555 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.242 - 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 555 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.242 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(555\)    =    \(3 \cdot 5 \cdot 37\)
Sign: $0.242 - 0.970i$
Analytic conductor: \(4.43169\)
Root analytic conductor: \(2.10515\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{555} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 555,\ (\ :1/2),\ 0.242 - 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.371025 + 0.289638i\)
\(L(\frac12)\) \(\approx\) \(0.371025 + 0.289638i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.173 - 0.984i)T \)
5 \( 1 + (0.766 - 0.642i)T \)
37 \( 1 + (1.87 + 5.78i)T \)
good2 \( 1 + (-0.374 + 2.12i)T + (-1.87 - 0.684i)T^{2} \)
7 \( 1 + (3.21 - 2.69i)T + (1.21 - 6.89i)T^{2} \)
11 \( 1 + (-0.363 + 0.629i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (4.15 + 1.51i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (5.79 - 2.10i)T + (13.0 - 10.9i)T^{2} \)
19 \( 1 + (-1.09 - 6.18i)T + (-17.8 + 6.49i)T^{2} \)
23 \( 1 + (-4.24 - 7.35i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.48 + 2.56i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 1.38T + 31T^{2} \)
41 \( 1 + (1.17 + 0.429i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + 1.80T + 43T^{2} \)
47 \( 1 + (0.0448 + 0.0776i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-8.39 - 7.04i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (9.08 + 7.61i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (4.32 + 1.57i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-3.29 + 2.76i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (-0.00767 - 0.0435i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + 7.75T + 73T^{2} \)
79 \( 1 + (-11.2 + 9.41i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (10.7 - 3.90i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + (-12.7 - 10.7i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (1.10 + 1.91i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.96628551807947320576838721229, −10.18029656015901167383667750243, −9.490932502759538251220675257020, −8.879479452491443929398350438155, −7.45228618409042918933082677236, −6.15406132542454928729668296099, −5.08007189767823624509452012739, −3.82471375201219673016431241207, −3.13742842433852273432386582962, −2.17801565405545713922873683333, 0.22789947066631321905705915663, 2.73986742347810344185465830187, 4.35320744619272458558005565089, 4.97998773804012686447791300106, 6.62530628228201620348302117380, 6.82124705733474137021573608901, 7.42873412063746157829548369229, 8.686814397587140563504837426664, 9.263706648042497875294572163323, 10.49941236840170780368543760466

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