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Properties

Label 2-504-63.5-c1-0-12
Degree $2$
Conductor $504$
Sign $0.599 + 0.800i$
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  + (−1.09 + 1.34i)3-s + (−0.271 − 0.469i)5-s + (1.60 − 2.10i)7-s + (−0.599 − 2.93i)9-s + (−0.666 − 0.384i)11-s + (−2.96 − 1.71i)13-s + (0.926 + 0.150i)15-s + (−3.23 − 5.60i)17-s + (5.60 + 3.23i)19-s + (1.05 + 4.45i)21-s + (0.100 − 0.0579i)23-s + (2.35 − 4.07i)25-s + (4.59 + 2.41i)27-s + (4.40 − 2.54i)29-s − 4.63i·31-s + ⋯
L(s)  = 1  + (−0.632 + 0.774i)3-s + (−0.121 − 0.209i)5-s + (0.607 − 0.794i)7-s + (−0.199 − 0.979i)9-s + (−0.200 − 0.115i)11-s + (−0.822 − 0.474i)13-s + (0.239 + 0.0389i)15-s + (−0.784 − 1.35i)17-s + (1.28 + 0.742i)19-s + (0.231 + 0.972i)21-s + (0.0209 − 0.0120i)23-s + (0.470 − 0.815i)25-s + (0.885 + 0.465i)27-s + (0.817 − 0.471i)29-s − 0.832i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.599 + 0.800i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.599 + 0.800i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.599 + 0.800i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.599 + 0.800i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.890716 - 0.445358i\)
\(L(\frac12)\) \(\approx\) \(0.890716 - 0.445358i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.09 - 1.34i)T \)
7 \( 1 + (-1.60 + 2.10i)T \)
good5 \( 1 + (0.271 + 0.469i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.666 + 0.384i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.96 + 1.71i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.23 + 5.60i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-5.60 - 3.23i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.100 + 0.0579i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.40 + 2.54i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 4.63iT - 31T^{2} \)
37 \( 1 + (-5.47 + 9.48i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.04 - 7.00i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.32 - 5.76i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 1.54T + 47T^{2} \)
53 \( 1 + (0.221 - 0.127i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 10.2T + 59T^{2} \)
61 \( 1 - 5.57iT - 61T^{2} \)
67 \( 1 + 3.28T + 67T^{2} \)
71 \( 1 + 5.67iT - 71T^{2} \)
73 \( 1 + (5.35 - 3.09i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 4.02T + 79T^{2} \)
83 \( 1 + (5.80 + 10.0i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (2.00 - 3.47i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-15.0 + 8.69i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.79399439386444333269050058863, −9.955591068669725269038760183503, −9.289651412746170326791789304823, −7.993737574704123516114018895496, −7.22058893137171496589475527780, −5.96478625942094837733030158115, −4.89737913011350059866757913334, −4.33678617070182797768240623597, −2.89363698594262383866529736236, −0.67521061778962391699496210073, 1.59891638019087857393595381486, 2.81139858186875936465973596935, 4.68689277350484836498663675438, 5.41185737959092313938887744648, 6.54948535791820967796415502165, 7.30341414459992383691808067567, 8.272690628669065394400283714151, 9.135911210992667117643883729067, 10.42494076158346280187414673617, 11.18885973978955778573021440443

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