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Properties

Label 2-1248-104.99-c1-0-26
Degree $2$
Conductor $1248$
Sign $-0.177 + 0.984i$
Analytic cond. $9.96533$
Root an. cond. $3.15679$
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  + 3-s + (0.273 − 0.273i)5-s + (−1.75 − 1.75i)7-s + 9-s + (3.22 − 3.22i)11-s + (−1.42 − 3.31i)13-s + (0.273 − 0.273i)15-s + 2.18i·17-s + (−5.33 − 5.33i)19-s + (−1.75 − 1.75i)21-s − 6.75·23-s + 4.85i·25-s + 27-s + 0.239i·29-s + (1.75 − 1.75i)31-s + ⋯
L(s)  = 1  + 0.577·3-s + (0.122 − 0.122i)5-s + (−0.665 − 0.665i)7-s + 0.333·9-s + (0.973 − 0.973i)11-s + (−0.396 − 0.918i)13-s + (0.0706 − 0.0706i)15-s + 0.530i·17-s + (−1.22 − 1.22i)19-s + (−0.383 − 0.383i)21-s − 1.40·23-s + 0.970i·25-s + 0.192·27-s + 0.0444i·29-s + (0.316 − 0.316i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.177 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.177 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1248\)    =    \(2^{5} \cdot 3 \cdot 13\)
Sign: $-0.177 + 0.984i$
Analytic conductor: \(9.96533\)
Root analytic conductor: \(3.15679\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1248} (463, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1248,\ (\ :1/2),\ -0.177 + 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.565036173\)
\(L(\frac12)\) \(\approx\) \(1.565036173\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
13 \( 1 + (1.42 + 3.31i)T \)
good5 \( 1 + (-0.273 + 0.273i)T - 5iT^{2} \)
7 \( 1 + (1.75 + 1.75i)T + 7iT^{2} \)
11 \( 1 + (-3.22 + 3.22i)T - 11iT^{2} \)
17 \( 1 - 2.18iT - 17T^{2} \)
19 \( 1 + (5.33 + 5.33i)T + 19iT^{2} \)
23 \( 1 + 6.75T + 23T^{2} \)
29 \( 1 - 0.239iT - 29T^{2} \)
31 \( 1 + (-1.75 + 1.75i)T - 31iT^{2} \)
37 \( 1 + (-3.35 - 3.35i)T + 37iT^{2} \)
41 \( 1 + (-1.29 - 1.29i)T + 41iT^{2} \)
43 \( 1 + 2.60iT - 43T^{2} \)
47 \( 1 + (7.61 + 7.61i)T + 47iT^{2} \)
53 \( 1 + 11.6iT - 53T^{2} \)
59 \( 1 + (-9.48 + 9.48i)T - 59iT^{2} \)
61 \( 1 + 1.47iT - 61T^{2} \)
67 \( 1 + (-7.93 - 7.93i)T + 67iT^{2} \)
71 \( 1 + (-5.30 + 5.30i)T - 71iT^{2} \)
73 \( 1 + (-3.04 + 3.04i)T - 73iT^{2} \)
79 \( 1 - 16.2iT - 79T^{2} \)
83 \( 1 + (-2.81 - 2.81i)T + 83iT^{2} \)
89 \( 1 + (0.702 - 0.702i)T - 89iT^{2} \)
97 \( 1 + (3.22 + 3.22i)T + 97iT^{2} \)
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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

−9.571474138898493132113705574602, −8.532751512029431111373136719642, −8.089502805125583446805754741104, −6.85747000476852071552020535563, −6.37346830717101702551619308842, −5.22389525498676572251573656981, −3.97360579817161395990065491166, −3.41687587729530770936710283837, −2.16047095255507582723144504942, −0.59191141535673513479111183590, 1.81758229639100712004057349858, 2.58309791960237006146422913414, 3.94653199285984113043267029760, 4.51187481952532356362740217658, 6.05896149457779568637563296343, 6.50500631678776378473679777914, 7.50001479029709585353322162666, 8.393811711786132004934106964532, 9.269689077584545567990525575723, 9.712625034642968866739151639363

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