Properties

Label 2-504-9.4-c1-0-6
Degree $2$
Conductor $504$
Sign $0.952 - 0.303i$
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  + (−1.30 − 1.14i)3-s + (−0.164 + 0.284i)5-s + (0.5 + 0.866i)7-s + (0.400 + 2.97i)9-s + (0.664 + 1.15i)11-s + (−1.53 + 2.66i)13-s + (0.539 − 0.183i)15-s + 7.35·17-s − 2.93·19-s + (0.335 − 1.69i)21-s + (3.34 − 5.79i)23-s + (2.44 + 4.23i)25-s + (2.86 − 4.33i)27-s + (3.88 + 6.72i)29-s + (1.63 − 2.83i)31-s + ⋯
L(s)  = 1  + (−0.752 − 0.658i)3-s + (−0.0735 + 0.127i)5-s + (0.188 + 0.327i)7-s + (0.133 + 0.991i)9-s + (0.200 + 0.347i)11-s + (−0.426 + 0.739i)13-s + (0.139 − 0.0474i)15-s + 1.78·17-s − 0.673·19-s + (0.0732 − 0.370i)21-s + (0.697 − 1.20i)23-s + (0.489 + 0.847i)25-s + (0.552 − 0.833i)27-s + (0.720 + 1.24i)29-s + (0.293 − 0.508i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 - 0.303i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.952 - 0.303i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.952 - 0.303i$
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: \((2,\ 504,\ (\ :1/2),\ 0.952 - 0.303i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.09820 + 0.170627i\)
\(L(\frac12)\) \(\approx\) \(1.09820 + 0.170627i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.30 + 1.14i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + (0.164 - 0.284i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.664 - 1.15i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.53 - 2.66i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 7.35T + 17T^{2} \)
19 \( 1 + 2.93T + 19T^{2} \)
23 \( 1 + (-3.34 + 5.79i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.88 - 6.72i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.63 + 2.83i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 0.329T + 37T^{2} \)
41 \( 1 + (0.135 - 0.234i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.48 - 9.49i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.571 + 0.989i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 6.42T + 53T^{2} \)
59 \( 1 + (0.372 - 0.644i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.42 + 7.65i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.28 - 7.42i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 1.60T + 71T^{2} \)
73 \( 1 + 13.4T + 73T^{2} \)
79 \( 1 + (-0.628 - 1.08i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.0316 - 0.0548i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 11.3T + 89T^{2} \)
97 \( 1 + (-5.51 - 9.55i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.02550704662745611441903363053, −10.27910229337915099364377496312, −9.183252685593243185611899636260, −8.109527719335923602726685289555, −7.19916472461921078201396894742, −6.44327077885485953208042260796, −5.37850851240329298770211146793, −4.48478535740684108563758304794, −2.77362272530837118889519092522, −1.32209211604700397364856339217, 0.876628592919545967116490300346, 3.11389040894688474593759631171, 4.21203916665755600646459862584, 5.26462876744088257218187087337, 6.00752989285024510795109340379, 7.21274867438789936318476390698, 8.188544555754332579208854213581, 9.279903581178897556857343105548, 10.24450132917148946864443081427, 10.64125613526712368171223513946

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