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Properties

Label 2-3584-3584.1301-c0-0-0
Degree $2$
Conductor $3584$
Sign $-0.482 - 0.876i$
Analytic cond. $1.78864$
Root an. cond. $1.33740$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.970 + 0.242i)2-s + (0.881 + 0.471i)4-s + (−0.998 − 0.0490i)7-s + (0.740 + 0.671i)8-s + (−0.336 + 0.941i)9-s + (−1.79 + 0.693i)11-s + (−0.956 − 0.290i)14-s + (0.555 + 0.831i)16-s + (−0.555 + 0.831i)18-s + (−1.91 + 0.235i)22-s + (0.829 − 0.207i)23-s + (−0.595 + 0.803i)25-s + (−0.857 − 0.514i)28-s + (1.37 + 1.30i)29-s + (0.336 + 0.941i)32-s + ⋯
L(s)  = 1  + (0.970 + 0.242i)2-s + (0.881 + 0.471i)4-s + (−0.998 − 0.0490i)7-s + (0.740 + 0.671i)8-s + (−0.336 + 0.941i)9-s + (−1.79 + 0.693i)11-s + (−0.956 − 0.290i)14-s + (0.555 + 0.831i)16-s + (−0.555 + 0.831i)18-s + (−1.91 + 0.235i)22-s + (0.829 − 0.207i)23-s + (−0.595 + 0.803i)25-s + (−0.857 − 0.514i)28-s + (1.37 + 1.30i)29-s + (0.336 + 0.941i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.482 - 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.482 - 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3584\)    =    \(2^{9} \cdot 7\)
Sign: $-0.482 - 0.876i$
Analytic conductor: \(1.78864\)
Root analytic conductor: \(1.33740\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3584} (1301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3584,\ (\ :0),\ -0.482 - 0.876i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.577095142\)
\(L(\frac12)\) \(\approx\) \(1.577095142\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.970 - 0.242i)T \)
7 \( 1 + (0.998 + 0.0490i)T \)
good3 \( 1 + (0.336 - 0.941i)T^{2} \)
5 \( 1 + (0.595 - 0.803i)T^{2} \)
11 \( 1 + (1.79 - 0.693i)T + (0.740 - 0.671i)T^{2} \)
13 \( 1 + (-0.989 - 0.146i)T^{2} \)
17 \( 1 + (-0.555 + 0.831i)T^{2} \)
19 \( 1 + (-0.857 - 0.514i)T^{2} \)
23 \( 1 + (-0.829 + 0.207i)T + (0.881 - 0.471i)T^{2} \)
29 \( 1 + (-1.37 - 1.30i)T + (0.0490 + 0.998i)T^{2} \)
31 \( 1 + (-0.382 + 0.923i)T^{2} \)
37 \( 1 + (1.47 - 1.14i)T + (0.242 - 0.970i)T^{2} \)
41 \( 1 + (0.290 - 0.956i)T^{2} \)
43 \( 1 + (0.341 + 1.96i)T + (-0.941 + 0.336i)T^{2} \)
47 \( 1 + (0.980 - 0.195i)T^{2} \)
53 \( 1 + (0.317 - 0.302i)T + (0.0490 - 0.998i)T^{2} \)
59 \( 1 + (0.989 - 0.146i)T^{2} \)
61 \( 1 + (-0.427 - 0.903i)T^{2} \)
67 \( 1 + (-0.431 - 1.92i)T + (-0.903 + 0.427i)T^{2} \)
71 \( 1 + (-0.661 + 1.39i)T + (-0.634 - 0.773i)T^{2} \)
73 \( 1 + (-0.995 + 0.0980i)T^{2} \)
79 \( 1 + (-0.186 - 0.226i)T + (-0.195 + 0.980i)T^{2} \)
83 \( 1 + (-0.242 - 0.970i)T^{2} \)
89 \( 1 + (-0.881 - 0.471i)T^{2} \)
97 \( 1 + (0.923 + 0.382i)T^{2} \)
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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

−8.704987525635320882972706300707, −8.121340114779254465559408846791, −7.13582750539664584816569840205, −6.96624047227712495954496655724, −5.73409409125785510614465755565, −5.19142886584110655807957395528, −4.65866351596137438248612282339, −3.37487887229487916814175132505, −2.83386150447037625962869884326, −1.95929246493552535381603649641, 0.59831962110227635630886694244, 2.37951686561981423855439147577, 3.02387674747169788571625381932, 3.63323680119738900760047607726, 4.70401976260474688050381484572, 5.51548157290521843514208866218, 6.16947888875786164832297104778, 6.67517407273350838805614856775, 7.67135389571873581568962040591, 8.383099380131060258099122245199

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