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Properties

Label 2-555-555.47-c1-0-8
Degree $2$
Conductor $555$
Sign $0.926 + 0.377i$
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  + (−0.578 − 2.16i)2-s + (−0.231 − 1.71i)3-s + (−2.60 + 1.50i)4-s + (2.08 + 0.814i)5-s + (−3.57 + 1.49i)6-s + (−1.33 + 4.97i)7-s + (1.58 + 1.58i)8-s + (−2.89 + 0.794i)9-s + (0.553 − 4.97i)10-s + 4.64i·11-s + (3.18 + 4.11i)12-s + (1.76 + 0.473i)13-s + 11.5·14-s + (0.915 − 3.76i)15-s + (−0.491 + 0.851i)16-s + (−0.363 − 1.35i)17-s + ⋯
L(s)  = 1  + (−0.409 − 1.52i)2-s + (−0.133 − 0.991i)3-s + (−1.30 + 0.751i)4-s + (0.931 + 0.364i)5-s + (−1.45 + 0.609i)6-s + (−0.504 + 1.88i)7-s + (0.561 + 0.561i)8-s + (−0.964 + 0.264i)9-s + (0.175 − 1.57i)10-s + 1.40i·11-s + (0.918 + 1.18i)12-s + (0.490 + 0.131i)13-s + 3.08·14-s + (0.236 − 0.971i)15-s + (−0.122 + 0.212i)16-s + (−0.0882 − 0.329i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 555 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.926 + 0.377i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 555 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.926 + 0.377i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(555\)    =    \(3 \cdot 5 \cdot 37\)
Sign: $0.926 + 0.377i$
Analytic conductor: \(4.43169\)
Root analytic conductor: \(2.10515\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{555} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 555,\ (\ :1/2),\ 0.926 + 0.377i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.840166 - 0.164508i\)
\(L(\frac12)\) \(\approx\) \(0.840166 - 0.164508i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.231 + 1.71i)T \)
5 \( 1 + (-2.08 - 0.814i)T \)
37 \( 1 + (1.19 + 5.96i)T \)
good2 \( 1 + (0.578 + 2.16i)T + (-1.73 + i)T^{2} \)
7 \( 1 + (1.33 - 4.97i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 - 4.64iT - 11T^{2} \)
13 \( 1 + (-1.76 - 0.473i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (0.363 + 1.35i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (2.85 - 1.64i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.68 - 2.68i)T + 23iT^{2} \)
29 \( 1 + 6.85T + 29T^{2} \)
31 \( 1 + 3.07T + 31T^{2} \)
41 \( 1 + (3.26 - 1.88i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (6.22 - 6.22i)T - 43iT^{2} \)
47 \( 1 + (-6.62 + 6.62i)T - 47iT^{2} \)
53 \( 1 + (-2.55 + 0.683i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-0.812 + 1.40i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.47 - 2.55i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.808 + 3.01i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-8.81 + 5.08i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.08 - 2.08i)T - 73iT^{2} \)
79 \( 1 + (-0.0724 + 0.0418i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.46 + 0.929i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (4.94 - 8.56i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.90 - 2.90i)T + 97iT^{2} \)
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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

−10.89067577514058510156992963450, −9.758418162129404635155689679154, −9.242994634735262599901582173237, −8.529021757455882860755198263695, −7.06145478344438612722616537911, −6.11946809474429727270059480989, −5.24336796454081530585060393590, −3.29896197512279522652057687038, −2.17356550909489667946292580943, −1.89242811227207683296734130170, 0.55636691311211457643534801360, 3.40032943944638312661515432259, 4.48595392736317046261505874819, 5.55069068842573753580254684073, 6.27302867259895601781229959437, 7.04626758140528142623851914686, 8.338601351407443017591963441734, 8.888888796260419793252027091406, 9.793283675050428932161286481623, 10.55796449465070300497263444373

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