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Properties

Label 2-1148-41.31-c1-0-16
Degree $2$
Conductor $1148$
Sign $-0.969 + 0.243i$
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  − 1.92i·3-s + (−0.139 − 0.428i)5-s + (0.587 + 0.809i)7-s − 0.709·9-s + (−2.10 − 0.683i)11-s + (−0.972 + 1.33i)13-s + (−0.825 + 0.268i)15-s + (−2.14 − 0.696i)17-s + (−5.11 − 7.04i)19-s + (1.55 − 1.13i)21-s + (−3.19 − 2.32i)23-s + (3.88 − 2.81i)25-s − 4.41i·27-s + (0.718 − 0.233i)29-s + (−1.81 + 5.58i)31-s + ⋯
L(s)  = 1  − 1.11i·3-s + (−0.0622 − 0.191i)5-s + (0.222 + 0.305i)7-s − 0.236·9-s + (−0.633 − 0.205i)11-s + (−0.269 + 0.371i)13-s + (−0.213 + 0.0692i)15-s + (−0.519 − 0.168i)17-s + (−1.17 − 1.61i)19-s + (0.340 − 0.247i)21-s + (−0.666 − 0.484i)23-s + (0.776 − 0.563i)25-s − 0.848i·27-s + (0.133 − 0.0433i)29-s + (−0.326 + 1.00i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 + 0.243i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.969 + 0.243i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $-0.969 + 0.243i$
Analytic conductor: \(9.16682\)
Root analytic conductor: \(3.02767\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1148,\ (\ :1/2),\ -0.969 + 0.243i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9558243192\)
\(L(\frac12)\) \(\approx\) \(0.9558243192\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.587 - 0.809i)T \)
41 \( 1 + (1.14 + 6.30i)T \)
good3 \( 1 + 1.92iT - 3T^{2} \)
5 \( 1 + (0.139 + 0.428i)T + (-4.04 + 2.93i)T^{2} \)
11 \( 1 + (2.10 + 0.683i)T + (8.89 + 6.46i)T^{2} \)
13 \( 1 + (0.972 - 1.33i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (2.14 + 0.696i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (5.11 + 7.04i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 + (3.19 + 2.32i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-0.718 + 0.233i)T + (23.4 - 17.0i)T^{2} \)
31 \( 1 + (1.81 - 5.58i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (2.05 + 6.33i)T + (-29.9 + 21.7i)T^{2} \)
43 \( 1 + (-0.280 - 0.203i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + (5.20 - 7.16i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (-1.15 + 0.375i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (2.93 + 2.13i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (7.32 - 5.31i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-11.0 + 3.60i)T + (54.2 - 39.3i)T^{2} \)
71 \( 1 + (1.00 + 0.326i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 - 9.55T + 73T^{2} \)
79 \( 1 + 6.59iT - 79T^{2} \)
83 \( 1 + 17.0T + 83T^{2} \)
89 \( 1 + (-0.654 - 0.900i)T + (-27.5 + 84.6i)T^{2} \)
97 \( 1 + (-14.0 + 4.57i)T + (78.4 - 57.0i)T^{2} \)
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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

−9.125500584488715278085609573014, −8.578885497448625385717962391746, −7.72701805427551380085815256665, −6.87388526032768578660321085012, −6.34809717516547828329043576680, −5.11138916332587607184103321886, −4.32784913101758866758055253144, −2.69523365092848720678486291360, −1.94651715079975185586431351036, −0.39253012315984274633986825983, 1.86245446674084959876951581256, 3.27848936877660141868847920492, 4.12239951987072715075365001525, 4.88534341893897839580525235552, 5.80469008523967455318587210534, 6.86096184533762865236445650987, 7.890608837217633462840150451782, 8.505836508167962866074431356410, 9.685745031670339938595458829662, 10.12682662354308473441869822334

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