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Properties

Label 2-495-45.38-c1-0-35
Degree $2$
Conductor $495$
Sign $0.624 + 0.780i$
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  + (−1.03 + 0.276i)2-s + (1.19 − 1.25i)3-s + (−0.741 + 0.427i)4-s + (1.51 − 1.64i)5-s + (−0.881 + 1.62i)6-s + (0.760 + 2.83i)7-s + (2.16 − 2.16i)8-s + (−0.165 − 2.99i)9-s + (−1.11 + 2.11i)10-s + (−0.866 − 0.5i)11-s + (−0.344 + 1.44i)12-s + (1.36 − 5.07i)13-s + (−1.57 − 2.72i)14-s + (−0.257 − 3.86i)15-s + (−0.777 + 1.34i)16-s + (1.26 + 1.26i)17-s + ⋯
L(s)  = 1  + (−0.730 + 0.195i)2-s + (0.687 − 0.726i)3-s + (−0.370 + 0.213i)4-s + (0.679 − 0.734i)5-s + (−0.359 + 0.665i)6-s + (0.287 + 1.07i)7-s + (0.763 − 0.763i)8-s + (−0.0551 − 0.998i)9-s + (−0.352 + 0.669i)10-s + (−0.261 − 0.150i)11-s + (−0.0993 + 0.416i)12-s + (0.377 − 1.40i)13-s + (−0.419 − 0.727i)14-s + (−0.0663 − 0.997i)15-s + (−0.194 + 0.336i)16-s + (0.306 + 0.306i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.624 + 0.780i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.624 + 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(495\)    =    \(3^{2} \cdot 5 \cdot 11\)
Sign: $0.624 + 0.780i$
Analytic conductor: \(3.95259\)
Root analytic conductor: \(1.98811\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{495} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 495,\ (\ :1/2),\ 0.624 + 0.780i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.12804 - 0.542055i\)
\(L(\frac12)\) \(\approx\) \(1.12804 - 0.542055i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.19 + 1.25i)T \)
5 \( 1 + (-1.51 + 1.64i)T \)
11 \( 1 + (0.866 + 0.5i)T \)
good2 \( 1 + (1.03 - 0.276i)T + (1.73 - i)T^{2} \)
7 \( 1 + (-0.760 - 2.83i)T + (-6.06 + 3.5i)T^{2} \)
13 \( 1 + (-1.36 + 5.07i)T + (-11.2 - 6.5i)T^{2} \)
17 \( 1 + (-1.26 - 1.26i)T + 17iT^{2} \)
19 \( 1 - 6.95iT - 19T^{2} \)
23 \( 1 + (-2.45 - 0.658i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (-4.34 + 7.52i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.67 + 6.36i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.250 + 0.250i)T - 37iT^{2} \)
41 \( 1 + (-7.57 + 4.37i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.08 - 1.36i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (-5.64 + 1.51i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (3.49 - 3.49i)T - 53iT^{2} \)
59 \( 1 + (-1.83 - 3.17i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.28 + 3.95i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (15.6 + 4.18i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 4.85iT - 71T^{2} \)
73 \( 1 + (-7.01 - 7.01i)T + 73iT^{2} \)
79 \( 1 + (-4.28 - 2.47i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.07 - 11.4i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + 2.56T + 89T^{2} \)
97 \( 1 + (-3.82 - 14.2i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.43586863212926938791830946520, −9.596741901833007523926053764057, −8.888703635853702614144641269219, −8.072606037046154364433338153667, −7.83912607133688473454076806080, −6.12770471836433573129557330386, −5.45408865163539927286297424243, −3.84890721045075879324833090894, −2.40420434529108657294265347261, −1.03042469107354474254840010266, 1.59098010659119895614665181288, 2.98113366755439782118116179868, 4.38160363879918129654538599065, 5.10578274778060875353383440246, 6.81344289654787261121778969957, 7.53320432439771267847786626713, 8.882833752739514261229401072688, 9.200955422363909449616570909512, 10.20323533451628627555445558414, 10.74131804929559464383922261127

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