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Properties

Label 2-1248-104.83-c1-0-2
Degree $2$
Conductor $1248$
Sign $-0.590 - 0.806i$
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 + (−1.61 − 1.61i)5-s + (−2.08 + 2.08i)7-s + 9-s + (0.0673 + 0.0673i)11-s + (−1.04 − 3.45i)13-s + (−1.61 − 1.61i)15-s + 7.18i·17-s + (−2.30 + 2.30i)19-s + (−2.08 + 2.08i)21-s − 2.00·23-s + 0.246i·25-s + 27-s + 4.37i·29-s + (2.08 + 2.08i)31-s + ⋯
L(s)  = 1  + 0.577·3-s + (−0.724 − 0.724i)5-s + (−0.786 + 0.786i)7-s + 0.333·9-s + (0.0203 + 0.0203i)11-s + (−0.288 − 0.957i)13-s + (−0.418 − 0.418i)15-s + 1.74i·17-s + (−0.528 + 0.528i)19-s + (−0.454 + 0.454i)21-s − 0.417·23-s + 0.0493i·25-s + 0.192·27-s + 0.811i·29-s + (0.373 + 0.373i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.590 - 0.806i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.590 - 0.806i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1248\)    =    \(2^{5} \cdot 3 \cdot 13\)
Sign: $-0.590 - 0.806i$
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: \((2,\ 1248,\ (\ :1/2),\ -0.590 - 0.806i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6442198147\)
\(L(\frac12)\) \(\approx\) \(0.6442198147\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.04 + 3.45i)T \)
good5 \( 1 + (1.61 + 1.61i)T + 5iT^{2} \)
7 \( 1 + (2.08 - 2.08i)T - 7iT^{2} \)
11 \( 1 + (-0.0673 - 0.0673i)T + 11iT^{2} \)
17 \( 1 - 7.18iT - 17T^{2} \)
19 \( 1 + (2.30 - 2.30i)T - 19iT^{2} \)
23 \( 1 + 2.00T + 23T^{2} \)
29 \( 1 - 4.37iT - 29T^{2} \)
31 \( 1 + (-2.08 - 2.08i)T + 31iT^{2} \)
37 \( 1 + (6.43 - 6.43i)T - 37iT^{2} \)
41 \( 1 + (7.94 - 7.94i)T - 41iT^{2} \)
43 \( 1 + 10.4iT - 43T^{2} \)
47 \( 1 + (5.31 - 5.31i)T - 47iT^{2} \)
53 \( 1 - 1.41iT - 53T^{2} \)
59 \( 1 + (-3.96 - 3.96i)T + 59iT^{2} \)
61 \( 1 + 1.94iT - 61T^{2} \)
67 \( 1 + (8.16 - 8.16i)T - 67iT^{2} \)
71 \( 1 + (0.00829 + 0.00829i)T + 71iT^{2} \)
73 \( 1 + (-0.419 - 0.419i)T + 73iT^{2} \)
79 \( 1 + 8.46iT - 79T^{2} \)
83 \( 1 + (-3.35 + 3.35i)T - 83iT^{2} \)
89 \( 1 + (-1.90 - 1.90i)T + 89iT^{2} \)
97 \( 1 + (-8.76 + 8.76i)T - 97iT^{2} \)
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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

−10.08693198225161628213150290217, −8.857423165769452638670495289153, −8.456517123892760708432640816165, −7.85857122205731094811516616604, −6.65336373528255276415154478806, −5.83335672286143081121888288259, −4.80176865611312246430409951861, −3.76668675846086294325660383956, −3.00631280901071008167824272838, −1.63180137768384688539778227248, 0.24259432575815827532574637396, 2.24139672438575923353517078911, 3.27127057323288429194105906654, 3.97577423316010067764920890484, 4.93250006744050408675474594481, 6.47769623322578170689933739832, 7.06533273290470253314268992391, 7.54535352778279716236378901130, 8.635626050475841249547642736442, 9.521712212110494480273410618408

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