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Properties

Label 2-1248-104.83-c1-0-7
Degree $2$
Conductor $1248$
Sign $-0.411 - 0.911i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.411 - 0.911i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.411 - 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1248\)    =    \(2^{5} \cdot 3 \cdot 13\)
Sign: $-0.411 - 0.911i$
Analytic conductor: \(9.96533\)
Root analytic conductor: \(3.15679\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1248} (655, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1248,\ (\ :1/2),\ -0.411 - 0.911i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.869080720\)
\(L(\frac12)\) \(\approx\) \(1.869080720\)
\(L(\frac{3}{2})\) 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

−9.874422085701125620216844714619, −9.326226807561658996348344622949, −8.526153125759623457974539287314, −7.32805059918313133883750811510, −6.56586606098372463866600159729, −5.94361670198691578636137303303, −5.13603312135259125731655564621, −3.28668951449670254489065645337, −2.80520295222023060561906627061, −2.08515917046041978592066661943, 0.68308781944206107030960918597, 2.06293831512490755930669996022, 3.02025480384066692997339680931, 4.53203721730973590854047192024, 4.95430655818301719301902583256, 6.12222706384929198581782002405, 7.01484635288665136791068016227, 7.82118585221197838834155744949, 8.858159135429468703668916619837, 9.590201401567458392902182194597

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