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Properties

Label 2-504-56.19-c1-0-15
Degree $2$
Conductor $504$
Sign $0.991 - 0.133i$
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  + (−1.41 − 0.0213i)2-s + (1.99 + 0.0602i)4-s + (1.14 − 1.97i)5-s + (1.95 + 1.78i)7-s + (−2.82 − 0.127i)8-s + (−1.65 + 2.76i)10-s + (2.60 + 4.50i)11-s + 1.44·13-s + (−2.72 − 2.56i)14-s + (3.99 + 0.240i)16-s + (−1.71 + 0.992i)17-s + (−4.27 − 2.47i)19-s + (2.39 − 3.88i)20-s + (−3.58 − 6.42i)22-s + (6.02 + 3.47i)23-s + ⋯
L(s)  = 1  + (−0.999 − 0.0150i)2-s + (0.999 + 0.0301i)4-s + (0.510 − 0.883i)5-s + (0.737 + 0.675i)7-s + (−0.998 − 0.0451i)8-s + (−0.523 + 0.875i)10-s + (0.784 + 1.35i)11-s + 0.400·13-s + (−0.727 − 0.686i)14-s + (0.998 + 0.0602i)16-s + (−0.416 + 0.240i)17-s + (−0.981 − 0.566i)19-s + (0.536 − 0.867i)20-s + (−0.763 − 1.36i)22-s + (1.25 + 0.725i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.133i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.133i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.991 - 0.133i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.991 - 0.133i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.13885 + 0.0764944i\)
\(L(\frac12)\) \(\approx\) \(1.13885 + 0.0764944i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 + 0.0213i)T \)
3 \( 1 \)
7 \( 1 + (-1.95 - 1.78i)T \)
good5 \( 1 + (-1.14 + 1.97i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.60 - 4.50i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 1.44T + 13T^{2} \)
17 \( 1 + (1.71 - 0.992i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.27 + 2.47i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-6.02 - 3.47i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 5.21iT - 29T^{2} \)
31 \( 1 + (1.69 + 2.93i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.53 + 1.46i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 6.20iT - 41T^{2} \)
43 \( 1 - 12.2T + 43T^{2} \)
47 \( 1 + (1.25 - 2.17i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.15 + 1.82i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-8.59 + 4.96i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.57 + 6.19i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.73 - 4.73i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 12.6iT - 71T^{2} \)
73 \( 1 + (9.13 - 5.27i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.38 + 2.53i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 0.265iT - 83T^{2} \)
89 \( 1 + (8.23 + 4.75i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 10.7iT - 97T^{2} \)
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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

−10.96511007586000548244907144298, −9.714469910584462910942009042196, −9.136062380353285109222656950756, −8.552403159027800711295199327469, −7.47787099151891176954066925918, −6.47828479620573198112222952171, −5.41634544780373056459784510239, −4.31113266973145766513157501702, −2.32472901696439538997617681643, −1.39747249945487909297257042229, 1.15151418266862816387344532504, 2.61591140505662103462648524014, 3.85837383699030897794703452231, 5.62317768288680008377854968667, 6.61746358277132699760486707182, 7.16385663457987465193539586475, 8.531451100848824528340320807912, 8.856702318913357363270767601437, 10.24182510305990645055184175105, 10.88153140335998893569716516927

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