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Properties

Label 2-3024-63.59-c1-0-12
Degree $2$
Conductor $3024$
Sign $-0.265 - 0.964i$
Analytic cond. $24.1467$
Root an. cond. $4.91393$
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  + 0.542·5-s + (2.62 − 0.340i)7-s + 0.769i·11-s + (2.96 + 1.71i)13-s + (−3.23 + 5.60i)17-s + (−5.60 + 3.23i)19-s + 0.115i·23-s − 4.70·25-s + (−4.40 + 2.54i)29-s + (−4.01 + 2.31i)31-s + (1.42 − 0.184i)35-s + (5.47 + 9.48i)37-s + (−4.04 + 7.00i)41-s + (−3.32 − 5.76i)43-s + (0.773 − 1.33i)47-s + ⋯
L(s)  = 1  + 0.242·5-s + (0.991 − 0.128i)7-s + 0.231i·11-s + (0.822 + 0.474i)13-s + (−0.784 + 1.35i)17-s + (−1.28 + 0.742i)19-s + 0.0241i·23-s − 0.941·25-s + (−0.817 + 0.471i)29-s + (−0.720 + 0.416i)31-s + (0.240 − 0.0311i)35-s + (0.900 + 1.55i)37-s + (−0.631 + 1.09i)41-s + (−0.507 − 0.878i)43-s + (0.112 − 0.195i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.265 - 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.265 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.265 - 0.964i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ -0.265 - 0.964i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.491647704\)
\(L(\frac12)\) \(\approx\) \(1.491647704\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.62 + 0.340i)T \)
good5 \( 1 - 0.542T + 5T^{2} \)
11 \( 1 - 0.769iT - 11T^{2} \)
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 \( 1 - 0.115iT - 23T^{2} \)
29 \( 1 + (4.40 - 2.54i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.01 - 2.31i)T + (15.5 - 26.8i)T^{2} \)
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 \( 1 + (-0.773 + 1.33i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.221 + 0.127i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.12 + 8.86i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.83 - 2.78i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.64 + 2.84i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 5.67iT - 71T^{2} \)
73 \( 1 + (5.35 + 3.09i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.01 + 3.48i)T + (-39.5 - 68.4i)T^{2} \)
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

−8.657485431565526603500403658963, −8.374149335367103421479642232331, −7.56034427193896066865194220127, −6.49378492099171742129320410208, −6.07033843597624239210357140440, −5.02478724040043679359048295654, −4.22402157501359283405211575703, −3.57464938542447342234642730038, −1.97052427902082224801437067497, −1.61089950483522323477197101167, 0.43467798720676333132200593639, 1.86690204355000802282623232215, 2.59760854260407208907345850809, 3.89272924156488476437333547843, 4.57411863496607325434194567304, 5.51895815370001045956178621495, 6.07466574123834641502532715697, 7.12477597471001521968420617418, 7.73190979125104795736112054801, 8.639159687272362328092868888262

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