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Properties

Label 2-495-45.34-c1-0-56
Degree $2$
Conductor $495$
Sign $-0.214 - 0.976i$
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  + (−1.18 − 0.681i)2-s + (1.20 − 1.24i)3-s + (−0.0706 − 0.122i)4-s + (−2.20 − 0.369i)5-s + (−2.27 + 0.648i)6-s + (−1.74 − 1.01i)7-s + 2.91i·8-s + (−0.0989 − 2.99i)9-s + (2.35 + 1.93i)10-s + (−0.5 + 0.866i)11-s + (−0.237 − 0.0594i)12-s + (−2.54 + 1.46i)13-s + (1.37 + 2.38i)14-s + (−3.11 + 2.30i)15-s + (1.84 − 3.20i)16-s + 0.804i·17-s + ⋯
L(s)  = 1  + (−0.834 − 0.482i)2-s + (0.695 − 0.718i)3-s + (−0.0353 − 0.0611i)4-s + (−0.986 − 0.165i)5-s + (−0.926 + 0.264i)6-s + (−0.661 − 0.381i)7-s + 1.03i·8-s + (−0.0329 − 0.999i)9-s + (0.743 + 0.613i)10-s + (−0.150 + 0.261i)11-s + (−0.0684 − 0.0171i)12-s + (−0.706 + 0.407i)13-s + (0.368 + 0.637i)14-s + (−0.804 + 0.593i)15-s + (0.462 − 0.800i)16-s + 0.195i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.214 - 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.214 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(495\)    =    \(3^{2} \cdot 5 \cdot 11\)
Sign: $-0.214 - 0.976i$
Analytic conductor: \(3.95259\)
Root analytic conductor: \(1.98811\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{495} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 495,\ (\ :1/2),\ -0.214 - 0.976i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0586722 + 0.0729518i\)
\(L(\frac12)\) \(\approx\) \(0.0586722 + 0.0729518i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.20 + 1.24i)T \)
5 \( 1 + (2.20 + 0.369i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (1.18 + 0.681i)T + (1 + 1.73i)T^{2} \)
7 \( 1 + (1.74 + 1.01i)T + (3.5 + 6.06i)T^{2} \)
13 \( 1 + (2.54 - 1.46i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 0.804iT - 17T^{2} \)
19 \( 1 - 3.27T + 19T^{2} \)
23 \( 1 + (6.06 - 3.50i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.30 - 5.72i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.0957 - 0.165i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 9.84iT - 37T^{2} \)
41 \( 1 + (0.767 + 1.32i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.03 + 0.600i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.461 - 0.266i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 10.9iT - 53T^{2} \)
59 \( 1 + (3.70 + 6.42i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.88 - 10.1i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.40 - 3.69i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 2.20T + 71T^{2} \)
73 \( 1 + 1.09iT - 73T^{2} \)
79 \( 1 + (1.63 - 2.83i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (6.67 + 3.85i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 12.5T + 89T^{2} \)
97 \( 1 + (6.79 + 3.92i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.09447545998810069494833933801, −9.361787477363637528554934241300, −8.646856904395361992099119075632, −7.59853750278109106543106499110, −7.19840888582561748843602394056, −5.68810012359050751980345494880, −4.18212308982450494327836202253, −3.02883696535568232753934756942, −1.63044480656076207027185609101, −0.06750934613524237836001716750, 2.84205484361498377329119380342, 3.70101620659068663880356386776, 4.77762161376481974011742691616, 6.31333852162151145141149262193, 7.53652393342406659915267682950, 8.018241045538159424500968492000, 8.800909870899521667537382038058, 9.752422978926533885211610481453, 10.18597965865191430309156734776, 11.47405812932642250304158891818

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