Properties

Label 2-495-45.32-c1-0-53
Degree $2$
Conductor $495$
Sign $0.243 + 0.969i$
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  + (1.67 + 0.448i)2-s + (−0.777 − 1.54i)3-s + (0.868 + 0.501i)4-s + (1.73 − 1.40i)5-s + (−0.607 − 2.93i)6-s + (0.209 − 0.783i)7-s + (−1.22 − 1.22i)8-s + (−1.79 + 2.40i)9-s + (3.53 − 1.57i)10-s + (0.866 − 0.5i)11-s + (0.100 − 1.73i)12-s + (−0.582 − 2.17i)13-s + (0.702 − 1.21i)14-s + (−3.53 − 1.59i)15-s + (−2.49 − 4.33i)16-s + (0.378 − 0.378i)17-s + ⋯
L(s)  = 1  + (1.18 + 0.317i)2-s + (−0.448 − 0.893i)3-s + (0.434 + 0.250i)4-s + (0.776 − 0.630i)5-s + (−0.247 − 1.19i)6-s + (0.0793 − 0.295i)7-s + (−0.431 − 0.431i)8-s + (−0.597 + 0.802i)9-s + (1.11 − 0.499i)10-s + (0.261 − 0.150i)11-s + (0.0291 − 0.500i)12-s + (−0.161 − 0.603i)13-s + (0.187 − 0.325i)14-s + (−0.911 − 0.411i)15-s + (−0.624 − 1.08i)16-s + (0.0918 − 0.0918i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.243 + 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.243 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(495\)    =    \(3^{2} \cdot 5 \cdot 11\)
Sign: $0.243 + 0.969i$
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: \((2,\ 495,\ (\ :1/2),\ 0.243 + 0.969i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.79005 - 1.39570i\)
\(L(\frac12)\) \(\approx\) \(1.79005 - 1.39570i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.777 + 1.54i)T \)
5 \( 1 + (-1.73 + 1.40i)T \)
11 \( 1 + (-0.866 + 0.5i)T \)
good2 \( 1 + (-1.67 - 0.448i)T + (1.73 + i)T^{2} \)
7 \( 1 + (-0.209 + 0.783i)T + (-6.06 - 3.5i)T^{2} \)
13 \( 1 + (0.582 + 2.17i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (-0.378 + 0.378i)T - 17iT^{2} \)
19 \( 1 - 3.79iT - 19T^{2} \)
23 \( 1 + (-3.83 + 1.02i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (1.22 + 2.12i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.81 - 3.14i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.527 - 0.527i)T + 37iT^{2} \)
41 \( 1 + (-6.28 - 3.62i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-8.54 - 2.29i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (9.28 + 2.48i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-9.15 - 9.15i)T + 53iT^{2} \)
59 \( 1 + (0.180 - 0.312i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.81 + 6.61i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.34 + 0.361i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 13.6iT - 71T^{2} \)
73 \( 1 + (-2.63 + 2.63i)T - 73iT^{2} \)
79 \( 1 + (8.95 - 5.16i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.35 - 16.2i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + 0.142T + 89T^{2} \)
97 \( 1 + (-3.44 + 12.8i)T + (-84.0 - 48.5i)T^{2} \)
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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

−11.00557551929234825713169392118, −9.928910747593549520199331019818, −8.888042936721964391910407731147, −7.76624459955111063964857504354, −6.74162038438947480362081529680, −5.86194958292885083047384605192, −5.33207927247324383312239657748, −4.26638935900550298793585627717, −2.73232994894667216811543174222, −1.07413622667665621218300836145, 2.35189444280487299312496570537, 3.41827605350118142305082511547, 4.46114834916851327348556977309, 5.35277476827255043096920962175, 6.07007926626361577268118655267, 7.06805853141420112341987577103, 8.914940644325594536263180555931, 9.380139831312111791425810646308, 10.54002915289806272021714313693, 11.25041155358675496702988359677

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