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Properties

Label 2-495-45.32-c1-0-47
Degree $2$
Conductor $495$
Sign $0.0239 + 0.999i$
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  + (−0.332 − 0.0890i)2-s + (1.73 + 0.00850i)3-s + (−1.62 − 0.940i)4-s + (0.380 − 2.20i)5-s + (−0.574 − 0.157i)6-s + (0.0988 − 0.368i)7-s + (0.944 + 0.944i)8-s + (2.99 + 0.0294i)9-s + (−0.322 + 0.698i)10-s + (−0.866 + 0.5i)11-s + (−2.81 − 1.64i)12-s + (−0.325 − 1.21i)13-s + (−0.0656 + 0.113i)14-s + (0.676 − 3.81i)15-s + (1.65 + 2.86i)16-s + (2.19 − 2.19i)17-s + ⋯
L(s)  = 1  + (−0.234 − 0.0629i)2-s + (0.999 + 0.00490i)3-s + (−0.814 − 0.470i)4-s + (0.169 − 0.985i)5-s + (−0.234 − 0.0641i)6-s + (0.0373 − 0.139i)7-s + (0.333 + 0.333i)8-s + (0.999 + 0.00981i)9-s + (−0.101 + 0.220i)10-s + (−0.261 + 0.150i)11-s + (−0.812 − 0.474i)12-s + (−0.0903 − 0.337i)13-s + (−0.0175 + 0.0303i)14-s + (0.174 − 0.984i)15-s + (0.413 + 0.715i)16-s + (0.533 − 0.533i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0239 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0239 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(495\)    =    \(3^{2} \cdot 5 \cdot 11\)
Sign: $0.0239 + 0.999i$
Analytic conductor: \(3.95259\)
Root analytic conductor: \(1.98811\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{495} (122, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 495,\ (\ :1/2),\ 0.0239 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03525 - 1.01079i\)
\(L(\frac12)\) \(\approx\) \(1.03525 - 1.01079i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.73 - 0.00850i)T \)
5 \( 1 + (-0.380 + 2.20i)T \)
11 \( 1 + (0.866 - 0.5i)T \)
good2 \( 1 + (0.332 + 0.0890i)T + (1.73 + i)T^{2} \)
7 \( 1 + (-0.0988 + 0.368i)T + (-6.06 - 3.5i)T^{2} \)
13 \( 1 + (0.325 + 1.21i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (-2.19 + 2.19i)T - 17iT^{2} \)
19 \( 1 + 6.92iT - 19T^{2} \)
23 \( 1 + (5.27 - 1.41i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (4.00 + 6.93i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.653 + 1.13i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.26 - 5.26i)T + 37iT^{2} \)
41 \( 1 + (-2.14 - 1.23i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.83 - 1.29i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (8.36 + 2.24i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-6.82 - 6.82i)T + 53iT^{2} \)
59 \( 1 + (5.18 - 8.97i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.67 - 9.83i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.28 + 1.41i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 7.99iT - 71T^{2} \)
73 \( 1 + (-7.01 + 7.01i)T - 73iT^{2} \)
79 \( 1 + (-6.63 + 3.83i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.38 + 8.90i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + 9.85T + 89T^{2} \)
97 \( 1 + (0.913 - 3.40i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.29879929933411690259094573024, −9.600033446128483607366964335719, −9.094086784539356213747669791476, −8.167444365926183637158854455106, −7.52073632622062917579239778328, −5.87731656352634704631523131979, −4.80784365004683851092866659682, −4.07836754133025727771335119117, −2.42718558215429840061299983192, −0.912662550966355856204817209635, 2.00070238464388523986511837516, 3.40507439712768559602842414930, 3.99958992226101860163116748528, 5.56821322915813399039217282430, 6.85809180937253593528462928111, 7.86484008198527946389683300955, 8.287118048523134247822249689022, 9.478495415254176670474890408696, 9.979903374179585823455055321212, 10.89090022819793141583059847328

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