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Properties

Label 2-504-168.125-c1-0-14
Degree $2$
Conductor $504$
Sign $0.950 - 0.311i$
Analytic cond. $4.02446$
Root an. cond. $2.00610$
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  + (−1.28 + 0.581i)2-s + (1.32 − 1.50i)4-s + 2.64·7-s + (−0.832 + 2.70i)8-s + 0.913·11-s + (−3.41 + 1.53i)14-s + (−0.5 − 3.96i)16-s + (−1.17 + 0.531i)22-s − 1.91i·23-s + 5·25-s + (3.50 − 3.96i)28-s + 6.06·29-s + (2.95 + 4.82i)32-s − 6i·37-s + 12i·43-s + (1.20 − 1.36i)44-s + ⋯
L(s)  = 1  + (−0.911 + 0.411i)2-s + (0.661 − 0.750i)4-s + 0.999·7-s + (−0.294 + 0.955i)8-s + 0.275·11-s + (−0.911 + 0.411i)14-s + (−0.125 − 0.992i)16-s + (−0.250 + 0.113i)22-s − 0.399i·23-s + 25-s + (0.661 − 0.749i)28-s + 1.12·29-s + (0.522 + 0.852i)32-s − 0.986i·37-s + 1.82i·43-s + (0.182 − 0.206i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.950 - 0.311i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.950 - 0.311i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.950 - 0.311i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.950 - 0.311i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.07118 + 0.171096i\)
\(L(\frac12)\) \(\approx\) \(1.07118 + 0.171096i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.28 - 0.581i)T \)
3 \( 1 \)
7 \( 1 - 2.64T \)
good5 \( 1 - 5T^{2} \)
11 \( 1 - 0.913T + 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + 1.91iT - 23T^{2} \)
29 \( 1 - 6.06T + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + 6iT - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 12iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 14.5T + 53T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 + 61T^{2} \)
67 \( 1 - 15.8iT - 67T^{2} \)
71 \( 1 + 15.0iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−10.83192798931718889962495406661, −10.06993027550696069293750224742, −8.973871949689509257862372736195, −8.368625657139003431004029685150, −7.46288601718935251352963114435, −6.57403746203360799620918346206, −5.48732399908494539999150986323, −4.45865256018905143772288406250, −2.59500870419687005975677730429, −1.18069103034148406961991913193, 1.20156025623210625971303187106, 2.53217338002399353849356481718, 3.91129891911160524558866837001, 5.16383690506183821092081631294, 6.57240314272062111390701825633, 7.44208977450434440879971063194, 8.402051862651192028166988629420, 8.955435145173243609392837734492, 10.12351276292914212768997091698, 10.74438284473618594366485628917

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