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Properties

Label 2-504-21.17-c1-0-6
Degree $2$
Conductor $504$
Sign $-0.484 + 0.875i$
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  + (0.144 + 0.250i)5-s + (−2.26 − 1.36i)7-s + (−5.23 − 3.01i)11-s − 5.46i·13-s + (−2.22 + 3.85i)17-s + (3.51 − 2.03i)19-s + (1.11 − 0.645i)23-s + (2.45 − 4.25i)25-s + 0.377i·29-s + (−3.09 − 1.78i)31-s + (0.0126 − 0.766i)35-s + (1.01 + 1.75i)37-s − 5.50·41-s − 6.45·43-s + (−5.38 − 9.33i)47-s + ⋯
L(s)  = 1  + (0.0647 + 0.112i)5-s + (−0.857 − 0.514i)7-s + (−1.57 − 0.910i)11-s − 1.51i·13-s + (−0.540 + 0.935i)17-s + (0.806 − 0.465i)19-s + (0.232 − 0.134i)23-s + (0.491 − 0.851i)25-s + 0.0701i·29-s + (−0.556 − 0.321i)31-s + (0.00214 − 0.129i)35-s + (0.166 + 0.289i)37-s − 0.859·41-s − 0.984·43-s + (−0.785 − 1.36i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.484 + 0.875i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.484 + 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.484 + 0.875i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ -0.484 + 0.875i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.391571 - 0.664106i\)
\(L(\frac12)\) \(\approx\) \(0.391571 - 0.664106i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + (2.26 + 1.36i)T \)
good5 \( 1 + (-0.144 - 0.250i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (5.23 + 3.01i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 5.46iT - 13T^{2} \)
17 \( 1 + (2.22 - 3.85i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.51 + 2.03i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.11 + 0.645i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 0.377iT - 29T^{2} \)
31 \( 1 + (3.09 + 1.78i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.01 - 1.75i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 5.50T + 41T^{2} \)
43 \( 1 + 6.45T + 43T^{2} \)
47 \( 1 + (5.38 + 9.33i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-9.77 - 5.64i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.790 - 1.36i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-9.54 + 5.51i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.04 - 3.53i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 0.410iT - 71T^{2} \)
73 \( 1 + (11.1 + 6.41i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.01 - 10.4i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 0.155T + 83T^{2} \)
89 \( 1 + (3.34 + 5.79i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 16.5iT - 97T^{2} \)
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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

−10.34989714142782084553471097403, −10.17780755587566058922082988263, −8.684791986033928929219239677910, −7.990861176907970156808333338290, −6.98057217393581745189815484387, −5.91002946102123096469461753141, −5.08622560635796506923297711093, −3.51735610120863732634141026167, −2.72109919361909436632843432086, −0.43319238979965936159683964967, 2.06992537019615752229144882400, 3.21199185118814053459691545840, 4.70389993101300037929616084056, 5.49597879545650043915060533698, 6.80188551837528380280907500965, 7.40315111348286988090542057309, 8.712387393215432896183402592268, 9.506819563814898216152597371345, 10.11840517477696196059262507743, 11.31292172288365942034794145556

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