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Properties

Label 2-460-92.91-c1-0-17
Degree $2$
Conductor $460$
Sign $0.992 + 0.118i$
Analytic cond. $3.67311$
Root an. cond. $1.91653$
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  + (0.341 − 1.37i)2-s + 2.41i·3-s + (−1.76 − 0.938i)4-s − i·5-s + (3.31 + 0.827i)6-s + 2.44·7-s + (−1.89 + 2.10i)8-s − 2.85·9-s + (−1.37 − 0.341i)10-s + 2.59·11-s + (2.26 − 4.27i)12-s − 0.482·13-s + (0.834 − 3.34i)14-s + 2.41·15-s + (2.23 + 3.31i)16-s + 1.74i·17-s + ⋯
L(s)  = 1  + (0.241 − 0.970i)2-s + 1.39i·3-s + (−0.883 − 0.469i)4-s − 0.447i·5-s + (1.35 + 0.337i)6-s + 0.922·7-s + (−0.668 + 0.743i)8-s − 0.950·9-s + (−0.433 − 0.108i)10-s + 0.780·11-s + (0.655 − 1.23i)12-s − 0.133·13-s + (0.223 − 0.895i)14-s + 0.624·15-s + (0.559 + 0.828i)16-s + 0.422i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.118i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 + 0.118i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $0.992 + 0.118i$
Analytic conductor: \(3.67311\)
Root analytic conductor: \(1.91653\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 460,\ (\ :1/2),\ 0.992 + 0.118i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.63229 - 0.0968192i\)
\(L(\frac12)\) \(\approx\) \(1.63229 - 0.0968192i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.341 + 1.37i)T \)
5 \( 1 + iT \)
23 \( 1 + (-1.73 - 4.47i)T \)
good3 \( 1 - 2.41iT - 3T^{2} \)
7 \( 1 - 2.44T + 7T^{2} \)
11 \( 1 - 2.59T + 11T^{2} \)
13 \( 1 + 0.482T + 13T^{2} \)
17 \( 1 - 1.74iT - 17T^{2} \)
19 \( 1 - 7.09T + 19T^{2} \)
29 \( 1 - 1.03T + 29T^{2} \)
31 \( 1 + 2.87iT - 31T^{2} \)
37 \( 1 + 6.76iT - 37T^{2} \)
41 \( 1 - 0.907T + 41T^{2} \)
43 \( 1 + 1.65T + 43T^{2} \)
47 \( 1 - 4.40iT - 47T^{2} \)
53 \( 1 + 1.89iT - 53T^{2} \)
59 \( 1 - 3.79iT - 59T^{2} \)
61 \( 1 + 8.17iT - 61T^{2} \)
67 \( 1 + 15.6T + 67T^{2} \)
71 \( 1 + 11.0iT - 71T^{2} \)
73 \( 1 - 5.06T + 73T^{2} \)
79 \( 1 + 9.79T + 79T^{2} \)
83 \( 1 + 13.5T + 83T^{2} \)
89 \( 1 - 2.55iT - 89T^{2} \)
97 \( 1 - 1.34iT - 97T^{2} \)
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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

−11.17312000343578571017391842486, −10.14079594668930130848583070569, −9.425215061299231402386638059188, −8.870207011120888811254119832428, −7.68008112312573202006808709725, −5.72847080997699389814067917334, −4.95559302491053354988752597501, −4.18676939939769494689987666024, −3.26817072522698663829527201136, −1.47412473703452916780408670371, 1.23167766715764376276459004797, 2.97941064763798143618027374833, 4.55446260988421730082884368824, 5.65194308018804149606843927730, 6.72194222053345102046243567710, 7.22156806256647565506942222873, 8.028150346290043866126398628514, 8.844075919532629950705551730306, 10.02441987459889367589444413869, 11.55379210157968032402262717310

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