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Properties

Label 2-495-5.4-c1-0-16
Degree $2$
Conductor $495$
Sign $-0.139 + 0.990i$
Analytic cond. $3.95259$
Root an. cond. $1.98811$
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  − 1.21i·2-s + 0.525·4-s + (−0.311 + 2.21i)5-s − 4.90i·7-s − 3.06i·8-s + (2.68 + 0.377i)10-s + 11-s + 4.14i·13-s − 5.95·14-s − 2.67·16-s − 5.33i·17-s + 5.18·19-s + (−0.163 + 1.16i)20-s − 1.21i·22-s − 4i·23-s + ⋯
L(s)  = 1  − 0.858i·2-s + 0.262·4-s + (−0.139 + 0.990i)5-s − 1.85i·7-s − 1.08i·8-s + (0.850 + 0.119i)10-s + 0.301·11-s + 1.15i·13-s − 1.59·14-s − 0.668·16-s − 1.29i·17-s + 1.18·19-s + (−0.0365 + 0.260i)20-s − 0.258i·22-s − 0.834i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.139 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.139 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(495\)    =    \(3^{2} \cdot 5 \cdot 11\)
Sign: $-0.139 + 0.990i$
Analytic conductor: \(3.95259\)
Root analytic conductor: \(1.98811\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{495} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 495,\ (\ :1/2),\ -0.139 + 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.04548 - 1.20264i\)
\(L(\frac12)\) \(\approx\) \(1.04548 - 1.20264i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (0.311 - 2.21i)T \)
11 \( 1 - T \)
good2 \( 1 + 1.21iT - 2T^{2} \)
7 \( 1 + 4.90iT - 7T^{2} \)
13 \( 1 - 4.14iT - 13T^{2} \)
17 \( 1 + 5.33iT - 17T^{2} \)
19 \( 1 - 5.18T + 19T^{2} \)
23 \( 1 + 4iT - 23T^{2} \)
29 \( 1 - 1.80T + 29T^{2} \)
31 \( 1 - 2.62T + 31T^{2} \)
37 \( 1 + 5.80iT - 37T^{2} \)
41 \( 1 + 1.80T + 41T^{2} \)
43 \( 1 - 4.90iT - 43T^{2} \)
47 \( 1 - 7.05iT - 47T^{2} \)
53 \( 1 - 7.18iT - 53T^{2} \)
59 \( 1 - 1.67T + 59T^{2} \)
61 \( 1 - 0.755T + 61T^{2} \)
67 \( 1 - 4.85iT - 67T^{2} \)
71 \( 1 + 0.428T + 71T^{2} \)
73 \( 1 - 12.7iT - 73T^{2} \)
79 \( 1 - 6.42T + 79T^{2} \)
83 \( 1 - 2.90iT - 83T^{2} \)
89 \( 1 - 0.622T + 89T^{2} \)
97 \( 1 - 2.75iT - 97T^{2} \)
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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

−10.84306213403104144283639403562, −10.04248303475353284942570567939, −9.420545976398668408759725922445, −7.62274185286319882362154301807, −7.06632272017166931071338438166, −6.45525201162838663534242453572, −4.49026405510141741049910465252, −3.66494917761977425547892052108, −2.63876732292421309958553325094, −1.04098515773574478893460464512, 1.82144530906181830592579048912, 3.23994909073607618468814771713, 5.10938188007806446358880434704, 5.56177907911676104225171996992, 6.37123313130301180507229747948, 7.79738351188505252179058426977, 8.369818503946964916411136064807, 9.044287788882676720046650371748, 10.12798936437513607452108808395, 11.53514482075853850240806666277

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