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Properties

Label 2-1248-104.77-c3-0-2
Degree $2$
Conductor $1248$
Sign $-0.200 + 0.979i$
Analytic cond. $73.6343$
Root an. cond. $8.58104$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3i·3-s − 3.15·5-s + 31.8i·7-s − 9·9-s − 30.3·11-s + (42.8 − 18.9i)13-s − 9.47i·15-s + 33.4·17-s − 32.4·19-s − 95.4·21-s − 92.1·23-s − 115.·25-s − 27i·27-s − 11.6i·29-s + 328. i·31-s + ⋯
L(s)  = 1  + 0.577i·3-s − 0.282·5-s + 1.71i·7-s − 0.333·9-s − 0.831·11-s + (0.914 − 0.403i)13-s − 0.163i·15-s + 0.477·17-s − 0.392·19-s − 0.992·21-s − 0.835·23-s − 0.920·25-s − 0.192i·27-s − 0.0748i·29-s + 1.90i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.200 + 0.979i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.200 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1248\)    =    \(2^{5} \cdot 3 \cdot 13\)
Sign: $-0.200 + 0.979i$
Analytic conductor: \(73.6343\)
Root analytic conductor: \(8.58104\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1248} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1248,\ (\ :3/2),\ -0.200 + 0.979i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.08235430543\)
\(L(\frac12)\) \(\approx\) \(0.08235430543\)
\(L(\frac{5}{2})\) 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 \( 1 - 3iT \)
13 \( 1 + (-42.8 + 18.9i)T \)
good5 \( 1 + 3.15T + 125T^{2} \)
7 \( 1 - 31.8iT - 343T^{2} \)
11 \( 1 + 30.3T + 1.33e3T^{2} \)
17 \( 1 - 33.4T + 4.91e3T^{2} \)
19 \( 1 + 32.4T + 6.85e3T^{2} \)
23 \( 1 + 92.1T + 1.21e4T^{2} \)
29 \( 1 + 11.6iT - 2.43e4T^{2} \)
31 \( 1 - 328. iT - 2.97e4T^{2} \)
37 \( 1 + 271.T + 5.06e4T^{2} \)
41 \( 1 + 153. iT - 6.89e4T^{2} \)
43 \( 1 - 26.1iT - 7.95e4T^{2} \)
47 \( 1 + 72.2iT - 1.03e5T^{2} \)
53 \( 1 - 666. iT - 1.48e5T^{2} \)
59 \( 1 - 512.T + 2.05e5T^{2} \)
61 \( 1 + 527. iT - 2.26e5T^{2} \)
67 \( 1 - 863.T + 3.00e5T^{2} \)
71 \( 1 + 810. iT - 3.57e5T^{2} \)
73 \( 1 + 157. iT - 3.89e5T^{2} \)
79 \( 1 + 796.T + 4.93e5T^{2} \)
83 \( 1 - 69.5T + 5.71e5T^{2} \)
89 \( 1 + 1.63e3iT - 7.04e5T^{2} \)
97 \( 1 - 904. iT - 9.12e5T^{2} \)
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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

−9.962084055213540151304392391873, −8.890280198230206398268857047606, −8.511686431622333057546195480429, −7.70582539088730586336233823321, −6.34914678866021635997646654625, −5.60041790637719464214136262537, −5.04204944120470622703244202410, −3.73784253702760131653900329199, −2.90848226194211351305569597926, −1.84809703285788656740905134911, 0.02129434148104198309148093941, 1.00475999452805029239208604565, 2.17612251072605210762996308906, 3.65921651643609766666807563918, 4.13572603967665867106166869655, 5.43359630461803732396004688856, 6.40345242859557979810625379340, 7.18543715214569961052560733681, 7.893825242638253707381311788198, 8.397934023317663431297535522826

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