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Properties

Label 2-555-185.184-c1-0-1
Degree $2$
Conductor $555$
Sign $-0.988 - 0.153i$
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  + 1.39·2-s + i·3-s − 0.0601·4-s + (−2.20 + 0.383i)5-s + 1.39i·6-s + 0.344i·7-s − 2.86·8-s − 9-s + (−3.06 + 0.534i)10-s − 1.00·11-s − 0.0601i·12-s − 4.37·13-s + 0.479i·14-s + (−0.383 − 2.20i)15-s − 3.87·16-s + 0.428·17-s + ⋯
L(s)  = 1  + 0.984·2-s + 0.577i·3-s − 0.0300·4-s + (−0.985 + 0.171i)5-s + 0.568i·6-s + 0.130i·7-s − 1.01·8-s − 0.333·9-s + (−0.970 + 0.169i)10-s − 0.301·11-s − 0.0173i·12-s − 1.21·13-s + 0.128i·14-s + (−0.0991 − 0.568i)15-s − 0.969·16-s + 0.103·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 555 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.153i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 555 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.988 - 0.153i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(555\)    =    \(3 \cdot 5 \cdot 37\)
Sign: $-0.988 - 0.153i$
Analytic conductor: \(4.43169\)
Root analytic conductor: \(2.10515\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{555} (184, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 555,\ (\ :1/2),\ -0.988 - 0.153i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0462560 + 0.599559i\)
\(L(\frac12)\) \(\approx\) \(0.0462560 + 0.599559i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - iT \)
5 \( 1 + (2.20 - 0.383i)T \)
37 \( 1 + (1.95 - 5.76i)T \)
good2 \( 1 - 1.39T + 2T^{2} \)
7 \( 1 - 0.344iT - 7T^{2} \)
11 \( 1 + 1.00T + 11T^{2} \)
13 \( 1 + 4.37T + 13T^{2} \)
17 \( 1 - 0.428T + 17T^{2} \)
19 \( 1 - 5.70iT - 19T^{2} \)
23 \( 1 + 2.35T + 23T^{2} \)
29 \( 1 + 7.62iT - 29T^{2} \)
31 \( 1 - 4.55iT - 31T^{2} \)
41 \( 1 - 2.50T + 41T^{2} \)
43 \( 1 + 8.58T + 43T^{2} \)
47 \( 1 - 7.01iT - 47T^{2} \)
53 \( 1 - 7.41iT - 53T^{2} \)
59 \( 1 + 3.94iT - 59T^{2} \)
61 \( 1 + 2.18iT - 61T^{2} \)
67 \( 1 + 10.2iT - 67T^{2} \)
71 \( 1 - 1.92T + 71T^{2} \)
73 \( 1 - 9.22iT - 73T^{2} \)
79 \( 1 - 14.3iT - 79T^{2} \)
83 \( 1 + 0.402iT - 83T^{2} \)
89 \( 1 + 7.34iT - 89T^{2} \)
97 \( 1 - 9.66T + 97T^{2} \)
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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

−11.47784848560652530272615275453, −10.31002862090802749796693656325, −9.597615452570149167453472679377, −8.436402961699564531808247393244, −7.69523463815087713353240471174, −6.41243660563096271516156972027, −5.34511630725501570226465613997, −4.52646572292310222378562775243, −3.73285619601974683249318277087, −2.73248413507161762858842931752, 0.24008559550917874361616843787, 2.55746815717391774153075380769, 3.66351638611514138647746751168, 4.72910813201760597578306717223, 5.41230293766040794856622102723, 6.79053987274219450313030956636, 7.45424977075146882008494044464, 8.508141283647585831074742395229, 9.318898781699514721664586865117, 10.59886158123675124975159783561

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