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Properties

Label 2-1248-104.99-c1-0-5
Degree $2$
Conductor $1248$
Sign $-0.812 - 0.583i$
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 + (−3.08 + 3.08i)5-s + (2.85 + 2.85i)7-s + 9-s + (−2.57 + 2.57i)11-s + (3.51 + 0.821i)13-s + (−3.08 + 3.08i)15-s − 1.07i·17-s + (−1.46 − 1.46i)19-s + (2.85 + 2.85i)21-s − 5.52·23-s − 14.0i·25-s + 27-s + 0.512i·29-s + (−2.85 + 2.85i)31-s + ⋯
L(s)  = 1  + 0.577·3-s + (−1.38 + 1.38i)5-s + (1.08 + 1.08i)7-s + 0.333·9-s + (−0.776 + 0.776i)11-s + (0.973 + 0.227i)13-s + (−0.797 + 0.797i)15-s − 0.259i·17-s + (−0.335 − 0.335i)19-s + (0.623 + 0.623i)21-s − 1.15·23-s − 2.81i·25-s + 0.192·27-s + 0.0951i·29-s + (−0.513 + 0.513i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.812 - 0.583i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.812 - 0.583i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1248\)    =    \(2^{5} \cdot 3 \cdot 13\)
Sign: $-0.812 - 0.583i$
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: \((2,\ 1248,\ (\ :1/2),\ -0.812 - 0.583i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.341005323\)
\(L(\frac12)\) \(\approx\) \(1.341005323\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.51 - 0.821i)T \)
good5 \( 1 + (3.08 - 3.08i)T - 5iT^{2} \)
7 \( 1 + (-2.85 - 2.85i)T + 7iT^{2} \)
11 \( 1 + (2.57 - 2.57i)T - 11iT^{2} \)
17 \( 1 + 1.07iT - 17T^{2} \)
19 \( 1 + (1.46 + 1.46i)T + 19iT^{2} \)
23 \( 1 + 5.52T + 23T^{2} \)
29 \( 1 - 0.512iT - 29T^{2} \)
31 \( 1 + (2.85 - 2.85i)T - 31iT^{2} \)
37 \( 1 + (1.97 + 1.97i)T + 37iT^{2} \)
41 \( 1 + (0.0336 + 0.0336i)T + 41iT^{2} \)
43 \( 1 - 3.49iT - 43T^{2} \)
47 \( 1 + (-6.45 - 6.45i)T + 47iT^{2} \)
53 \( 1 + 8.04iT - 53T^{2} \)
59 \( 1 + (1.23 - 1.23i)T - 59iT^{2} \)
61 \( 1 - 4.66iT - 61T^{2} \)
67 \( 1 + (2.03 + 2.03i)T + 67iT^{2} \)
71 \( 1 + (5.60 - 5.60i)T - 71iT^{2} \)
73 \( 1 + (5.73 - 5.73i)T - 73iT^{2} \)
79 \( 1 - 6.65iT - 79T^{2} \)
83 \( 1 + (0.153 + 0.153i)T + 83iT^{2} \)
89 \( 1 + (-11.3 + 11.3i)T - 89iT^{2} \)
97 \( 1 + (-8.80 - 8.80i)T + 97iT^{2} \)
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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

−10.20988830663595697380612455644, −8.954719232778868620866603971063, −8.253480536821231003856149717916, −7.69834384187116942127262421029, −6.99689083616633713195759703803, −5.93970859630655537508594181845, −4.70376900204349691417341278456, −3.88036322518491847351218324916, −2.84464725537300164155752405792, −2.03569372879550187150066777048, 0.53552814478091073696272903568, 1.64319208669632777380978361621, 3.54379190681336883335078607667, 4.06939394706894659724276622354, 4.84273173712223037866160342832, 5.85586986093740030507870487663, 7.45282364275551441875673024104, 7.87787998917688366297920896828, 8.419607700345052913799835805150, 8.960848220696671421985867209040

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