Properties

Label 2-504-7.4-c1-0-0
Degree $2$
Conductor $504$
Sign $-0.786 - 0.617i$
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  + (−2.13 − 3.70i)5-s + (−1.5 + 2.17i)7-s + (−2.13 + 3.70i)11-s − 1.27·13-s + (−2 + 3.46i)17-s + (−0.637 − 1.10i)19-s + (2 + 3.46i)23-s + (−6.63 + 11.4i)25-s + 2.27·29-s + (0.5 − 0.866i)31-s + (11.2 + 0.894i)35-s + (−2.63 − 4.56i)37-s − 10.5·41-s − 7.27·43-s + (−3 − 5.19i)47-s + ⋯
L(s)  = 1  + (−0.955 − 1.65i)5-s + (−0.566 + 0.823i)7-s + (−0.644 + 1.11i)11-s − 0.353·13-s + (−0.485 + 0.840i)17-s + (−0.146 − 0.253i)19-s + (0.417 + 0.722i)23-s + (−1.32 + 2.29i)25-s + 0.422·29-s + (0.0898 − 0.155i)31-s + (1.90 + 0.151i)35-s + (−0.433 − 0.751i)37-s − 1.64·41-s − 1.10·43-s + (−0.437 − 0.757i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.786 - 0.617i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.786 - 0.617i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.786 - 0.617i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ -0.786 - 0.617i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0549703 + 0.158885i\)
\(L(\frac12)\) \(\approx\) \(0.0549703 + 0.158885i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.5 - 2.17i)T \)
good5 \( 1 + (2.13 + 3.70i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.13 - 3.70i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 1.27T + 13T^{2} \)
17 \( 1 + (2 - 3.46i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.637 + 1.10i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2 - 3.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 2.27T + 29T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.63 + 4.56i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 10.5T + 41T^{2} \)
43 \( 1 + 7.27T + 43T^{2} \)
47 \( 1 + (3 + 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.862 + 1.49i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.13 + 5.43i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5 + 8.66i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.63 - 6.30i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 2T + 71T^{2} \)
73 \( 1 + (1.63 - 2.83i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.77 - 3.07i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 0.274T + 83T^{2} \)
89 \( 1 + (-2.27 - 3.94i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 16.2T + 97T^{2} \)
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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

−11.61473974535710019213498829638, −10.22800330923716578642442510122, −9.324359183546480807713480290607, −8.608746267605783828483976256724, −7.88110030602600671525786931753, −6.78577233948623627012317825020, −5.33106496025951954773054567490, −4.77957998509779172005242921213, −3.60014165387891517542180192832, −1.90612425236773959998685281957, 0.095151750686273758143060150562, 2.85189444471231326359137951074, 3.39110734290764065388311607061, 4.65429732110921560121921876757, 6.27557007422861267016905538631, 6.92536146533550938377626533515, 7.67319735820688875431296218028, 8.612467438850165086547316405815, 10.12042018597056303652459918391, 10.52887493167027224065377445609

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