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Properties

Label 2-495-45.4-c1-0-44
Degree $2$
Conductor $495$
Sign $0.946 + 0.324i$
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  + (2.08 − 1.20i)2-s + (1.33 + 1.10i)3-s + (1.91 − 3.30i)4-s + (−0.371 + 2.20i)5-s + (4.11 + 0.702i)6-s + (0.128 − 0.0743i)7-s − 4.39i·8-s + (0.554 + 2.94i)9-s + (1.88 + 5.05i)10-s + (−0.5 − 0.866i)11-s + (6.20 − 2.29i)12-s + (−0.501 − 0.289i)13-s + (0.179 − 0.310i)14-s + (−2.93 + 2.52i)15-s + (−1.48 − 2.56i)16-s − 6.95i·17-s + ⋯
L(s)  = 1  + (1.47 − 0.853i)2-s + (0.769 + 0.638i)3-s + (0.955 − 1.65i)4-s + (−0.166 + 0.986i)5-s + (1.68 + 0.286i)6-s + (0.0486 − 0.0281i)7-s − 1.55i·8-s + (0.184 + 0.982i)9-s + (0.595 + 1.59i)10-s + (−0.150 − 0.261i)11-s + (1.79 − 0.663i)12-s + (−0.139 − 0.0803i)13-s + (0.0479 − 0.0830i)14-s + (−0.757 + 0.652i)15-s + (−0.370 − 0.641i)16-s − 1.68i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.946 + 0.324i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.946 + 0.324i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(495\)    =    \(3^{2} \cdot 5 \cdot 11\)
Sign: $0.946 + 0.324i$
Analytic conductor: \(3.95259\)
Root analytic conductor: \(1.98811\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{495} (364, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 495,\ (\ :1/2),\ 0.946 + 0.324i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.67419 - 0.611758i\)
\(L(\frac12)\) \(\approx\) \(3.67419 - 0.611758i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.33 - 1.10i)T \)
5 \( 1 + (0.371 - 2.20i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (-2.08 + 1.20i)T + (1 - 1.73i)T^{2} \)
7 \( 1 + (-0.128 + 0.0743i)T + (3.5 - 6.06i)T^{2} \)
13 \( 1 + (0.501 + 0.289i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 6.95iT - 17T^{2} \)
19 \( 1 + 4.08T + 19T^{2} \)
23 \( 1 + (-0.856 - 0.494i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.73 + 4.73i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.72 + 4.72i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 1.78iT - 37T^{2} \)
41 \( 1 + (-1.88 + 3.25i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.14 - 2.97i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.94 - 1.70i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 8.91iT - 53T^{2} \)
59 \( 1 + (-6.52 + 11.3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.48 - 6.04i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-12.8 - 7.42i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 6.25T + 71T^{2} \)
73 \( 1 + 0.278iT - 73T^{2} \)
79 \( 1 + (1.98 + 3.43i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (8.47 - 4.89i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 5.10T + 89T^{2} \)
97 \( 1 + (11.7 - 6.78i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.22187041826618647955563219167, −10.22164047616391518824924389576, −9.578055118411576638027145029001, −8.166154020018723016769738683626, −7.08281715146725028116997769501, −5.90485905773328228378325571814, −4.79612458376157523237524205213, −3.95769717085525913120588466168, −2.95799582101134866780301323402, −2.32954393948968983109615897685, 1.88000133236918330319325285715, 3.45740601671403539533946127497, 4.28235617272233357223159998096, 5.31294158041662328442754909504, 6.35275938522115304367549900972, 7.13340239862266765115222365755, 8.236357051261143130540468171366, 8.601961585706231613505284953461, 10.00303188212493464902221851803, 11.50866847307925953670400693346

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