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Properties

Label 2-1288-161.68-c1-0-15
Degree $2$
Conductor $1288$
Sign $-0.524 - 0.851i$
Analytic cond. $10.2847$
Root an. cond. $3.20698$
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  + (0.124 − 0.0720i)3-s + (−2.06 + 3.57i)5-s + (2.62 + 0.294i)7-s + (−1.48 + 2.58i)9-s + (3.61 − 2.08i)11-s + 2.51i·13-s + 0.595i·15-s + (3.70 + 6.41i)17-s + (2.97 − 5.15i)19-s + (0.349 − 0.152i)21-s + (−3.40 − 3.37i)23-s + (−6.03 − 10.4i)25-s + 0.861i·27-s − 4.39·29-s + (−5.06 + 2.92i)31-s + ⋯
L(s)  = 1  + (0.0720 − 0.0415i)3-s + (−0.923 + 1.60i)5-s + (0.993 + 0.111i)7-s + (−0.496 + 0.860i)9-s + (1.09 − 0.630i)11-s + 0.697i·13-s + 0.153i·15-s + (0.898 + 1.55i)17-s + (0.682 − 1.18i)19-s + (0.0761 − 0.0333i)21-s + (−0.710 − 0.703i)23-s + (−1.20 − 2.09i)25-s + 0.165i·27-s − 0.815·29-s + (−0.909 + 0.525i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.524 - 0.851i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.524 - 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1288\)    =    \(2^{3} \cdot 7 \cdot 23\)
Sign: $-0.524 - 0.851i$
Analytic conductor: \(10.2847\)
Root analytic conductor: \(3.20698\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1288} (873, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1288,\ (\ :1/2),\ -0.524 - 0.851i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.422619023\)
\(L(\frac12)\) \(\approx\) \(1.422619023\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (-2.62 - 0.294i)T \)
23 \( 1 + (3.40 + 3.37i)T \)
good3 \( 1 + (-0.124 + 0.0720i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (2.06 - 3.57i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-3.61 + 2.08i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 2.51iT - 13T^{2} \)
17 \( 1 + (-3.70 - 6.41i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.97 + 5.15i)T + (-9.5 - 16.4i)T^{2} \)
29 \( 1 + 4.39T + 29T^{2} \)
31 \( 1 + (5.06 - 2.92i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.83 - 2.21i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 8.67iT - 41T^{2} \)
43 \( 1 - 8.08iT - 43T^{2} \)
47 \( 1 + (2.84 + 1.64i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (11.3 - 6.56i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.29 + 1.32i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.49 + 7.77i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.47 + 3.16i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.32T + 71T^{2} \)
73 \( 1 + (3.85 - 2.22i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (10.7 + 6.21i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 5.46T + 83T^{2} \)
89 \( 1 + (0.792 - 1.37i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 4.06T + 97T^{2} \)
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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

−10.13399199279737194577120864189, −8.976915911964143741902047971816, −8.031405281875927374383079111544, −7.71230217184180279287641652157, −6.64816434920476479209482013750, −5.99540457499847430978583297223, −4.68906056873702712662218253242, −3.76479900593809441469899404358, −2.92108565042728907003284130045, −1.69871116204053580331754460187, 0.63096791417305266792777536058, 1.62496622114763782949732810544, 3.58692436591213489447996671482, 4.08348041676401697913110209350, 5.24684577360450681058690765367, 5.64149383922298221390078815687, 7.36054683067944237914112485257, 7.72583472644542142021223064027, 8.616332678846695654326168837657, 9.296684925393545279755260396475

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