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Properties

Label 2-504-9.4-c1-0-2
Degree $2$
Conductor $504$
Sign $-0.966 + 0.256i$
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  + (−1.17 + 1.27i)3-s + (−2.19 + 3.79i)5-s + (0.5 + 0.866i)7-s + (−0.253 − 2.98i)9-s + (2.69 + 4.66i)11-s + (1.27 − 2.20i)13-s + (−2.27 − 7.23i)15-s − 2.58·17-s − 6.72·19-s + (−1.69 − 0.377i)21-s + (0.400 − 0.693i)23-s + (−7.09 − 12.2i)25-s + (4.10 + 3.18i)27-s + (−1.87 − 3.24i)29-s + (−1.69 + 2.93i)31-s + ⋯
L(s)  = 1  + (−0.676 + 0.736i)3-s + (−0.979 + 1.69i)5-s + (0.188 + 0.327i)7-s + (−0.0843 − 0.996i)9-s + (0.811 + 1.40i)11-s + (0.352 − 0.610i)13-s + (−0.586 − 1.86i)15-s − 0.625·17-s − 1.54·19-s + (−0.368 − 0.0823i)21-s + (0.0834 − 0.144i)23-s + (−1.41 − 2.45i)25-s + (0.790 + 0.612i)27-s + (−0.347 − 0.601i)29-s + (−0.304 + 0.527i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.966 + 0.256i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.966 + 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.966 + 0.256i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ -0.966 + 0.256i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0779544 - 0.598526i\)
\(L(\frac12)\) \(\approx\) \(0.0779544 - 0.598526i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.17 - 1.27i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + (2.19 - 3.79i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.69 - 4.66i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.27 + 2.20i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 2.58T + 17T^{2} \)
19 \( 1 + 6.72T + 19T^{2} \)
23 \( 1 + (-0.400 + 0.693i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.87 + 3.24i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.69 - 2.93i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 4.38T + 37T^{2} \)
41 \( 1 + (-3.19 + 5.53i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.381 - 0.661i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.13 - 7.16i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 4.94T + 53T^{2} \)
59 \( 1 + (2.78 - 4.82i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.14 - 7.17i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.946 + 1.63i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 1.34T + 71T^{2} \)
73 \( 1 + 8.65T + 73T^{2} \)
79 \( 1 + (-6.64 - 11.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.86 + 3.22i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 6.99T + 89T^{2} \)
97 \( 1 + (1.48 + 2.56i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.16522881995887111335275375979, −10.71353295787431056496988953782, −9.921454157300158929427951395923, −8.824669055174744341239228416443, −7.58953303249438086718114520130, −6.72949103176499848552626328448, −6.06295412946230036477552407578, −4.44618700599700940259150369698, −3.87163834180426580399237151758, −2.50829893564676645055763823502, 0.40458381859392318503843861753, 1.57363367894288468749480930427, 3.89285742028325132963814252311, 4.64248421272949003693282987290, 5.77803057306812573962581316579, 6.69899470005277418492036139585, 7.901594269668197815778614975950, 8.540916957973147910665763392034, 9.186873742675486898197661499308, 10.97401773906749270229113539766

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