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Properties

Label 2-455-91.74-c1-0-12
Degree $2$
Conductor $455$
Sign $0.918 - 0.396i$
Analytic cond. $3.63319$
Root an. cond. $1.90609$
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  − 0.460·2-s + (0.706 + 1.22i)3-s − 1.78·4-s + (−0.5 − 0.866i)5-s + (−0.325 − 0.563i)6-s + (−1.42 − 2.22i)7-s + 1.74·8-s + (0.500 − 0.866i)9-s + (0.230 + 0.398i)10-s + (2.79 + 4.84i)11-s + (−1.26 − 2.18i)12-s + (2.30 + 2.77i)13-s + (0.656 + 1.02i)14-s + (0.706 − 1.22i)15-s + 2.77·16-s + 1.65·17-s + ⋯
L(s)  = 1  − 0.325·2-s + (0.408 + 0.706i)3-s − 0.894·4-s + (−0.223 − 0.387i)5-s + (−0.132 − 0.230i)6-s + (−0.538 − 0.842i)7-s + 0.616·8-s + (0.166 − 0.288i)9-s + (0.0727 + 0.126i)10-s + (0.844 + 1.46i)11-s + (−0.364 − 0.632i)12-s + (0.639 + 0.769i)13-s + (0.175 + 0.274i)14-s + (0.182 − 0.316i)15-s + 0.693·16-s + 0.400·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 455 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.918 - 0.396i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 455 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.918 - 0.396i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(455\)    =    \(5 \cdot 7 \cdot 13\)
Sign: $0.918 - 0.396i$
Analytic conductor: \(3.63319\)
Root analytic conductor: \(1.90609\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{455} (256, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 455,\ (\ :1/2),\ 0.918 - 0.396i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10691 + 0.228550i\)
\(L(\frac12)\) \(\approx\) \(1.10691 + 0.228550i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (1.42 + 2.22i)T \)
13 \( 1 + (-2.30 - 2.77i)T \)
good2 \( 1 + 0.460T + 2T^{2} \)
3 \( 1 + (-0.706 - 1.22i)T + (-1.5 + 2.59i)T^{2} \)
11 \( 1 + (-2.79 - 4.84i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 - 1.65T + 17T^{2} \)
19 \( 1 + (-2.66 + 4.61i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 3.25T + 23T^{2} \)
29 \( 1 + (3.44 - 5.97i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.23 - 3.86i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 9.65T + 37T^{2} \)
41 \( 1 + (-3.14 + 5.45i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.93 + 5.08i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.473 + 0.819i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.83 + 3.18i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 11.2T + 59T^{2} \)
61 \( 1 + (5.30 - 9.18i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.10 + 10.5i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (4.47 + 7.74i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-0.350 + 0.606i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.90 - 6.76i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 10.4T + 83T^{2} \)
89 \( 1 - 3.34T + 89T^{2} \)
97 \( 1 + (1.23 + 2.13i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.82998278388020554920566661332, −9.926342793130113575134806892615, −9.194155914985718000463140376800, −8.988346275188897468982484579539, −7.46343905492170681900129602984, −6.77518042629424215980344764052, −4.99004050590436716606240872991, −4.20142233051294578645452384016, −3.55444780541652955487450621330, −1.16829061664861066625749572123, 1.08184757361793345235506730042, 2.92773318514448351256091008512, 3.88778855475072392446166383211, 5.56899692139138391045792421199, 6.26988232111585550392199121962, 7.77582949773619871015102734162, 8.175646864034613135402752815172, 9.120299924153603197148084674874, 9.897903906708706237189864864554, 11.02663041772321014250308217233

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