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Properties

Label 2-1148-7.2-c1-0-13
Degree $2$
Conductor $1148$
Sign $-0.635 + 0.771i$
Analytic cond. $9.16682$
Root an. cond. $3.02767$
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  + (−1.62 − 2.80i)3-s + (−0.748 + 1.29i)5-s + (2.53 − 0.768i)7-s + (−3.75 + 6.50i)9-s + (1.30 + 2.25i)11-s + 2.15·13-s + 4.85·15-s + (−3.90 − 6.77i)17-s + (2.49 − 4.32i)19-s + (−6.26 − 5.86i)21-s + (2.50 − 4.33i)23-s + (1.37 + 2.38i)25-s + 14.6·27-s + 0.404·29-s + (−0.566 − 0.981i)31-s + ⋯
L(s)  = 1  + (−0.935 − 1.62i)3-s + (−0.334 + 0.579i)5-s + (0.956 − 0.290i)7-s + (−1.25 + 2.16i)9-s + (0.392 + 0.679i)11-s + 0.598·13-s + 1.25·15-s + (−0.948 − 1.64i)17-s + (0.572 − 0.991i)19-s + (−1.36 − 1.27i)21-s + (0.521 − 0.902i)23-s + (0.275 + 0.477i)25-s + 2.81·27-s + 0.0750·29-s + (−0.101 − 0.176i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.635 + 0.771i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.635 + 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $-0.635 + 0.771i$
Analytic conductor: \(9.16682\)
Root analytic conductor: \(3.02767\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (821, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1148,\ (\ :1/2),\ -0.635 + 0.771i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.050939886\)
\(L(\frac12)\) \(\approx\) \(1.050939886\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.53 + 0.768i)T \)
41 \( 1 + T \)
good3 \( 1 + (1.62 + 2.80i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.748 - 1.29i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.30 - 2.25i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 2.15T + 13T^{2} \)
17 \( 1 + (3.90 + 6.77i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.49 + 4.32i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.50 + 4.33i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 0.404T + 29T^{2} \)
31 \( 1 + (0.566 + 0.981i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.838 + 1.45i)T + (-18.5 - 32.0i)T^{2} \)
43 \( 1 + 0.0986T + 43T^{2} \)
47 \( 1 + (3.36 - 5.82i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.51 + 11.2i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (7.59 + 13.1i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.936 + 1.62i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.30 - 3.98i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 1.21T + 71T^{2} \)
73 \( 1 + (1.58 + 2.73i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.28 - 10.8i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 12.0T + 83T^{2} \)
89 \( 1 + (6.16 - 10.6i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 16.3T + 97T^{2} \)
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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

−9.437593738230814293192687840278, −8.382703216948074689961926666219, −7.54729781668934822529775061352, −6.92753415759292963095832479823, −6.57398141790603914036757688888, −5.22440997333571391343374781073, −4.63618863158311778045145129061, −2.84562957079276512159535449404, −1.77850200288262292593849521217, −0.58958682243285263881789858415, 1.29968101009468161387681951022, 3.42537999796629355070447736039, 4.16243537380889480343404442820, 4.85354806729889416730709618097, 5.75288129855620498666665617187, 6.27262250715466120694203443745, 7.927430628898147340248291261857, 8.781522892782947746981244613510, 9.122010994631054213617357427299, 10.43883870153854153191084276945

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