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Properties

Label 2-1148-287.163-c1-0-24
Degree $2$
Conductor $1148$
Sign $0.376 + 0.926i$
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  + (2.89 − 1.67i)3-s + (1.42 − 2.47i)5-s + (1.36 + 2.26i)7-s + (4.08 − 7.07i)9-s + (−1.10 + 0.636i)11-s + 3.41i·13-s − 9.54i·15-s + (1.97 − 1.13i)17-s + (0.851 + 0.491i)19-s + (7.73 + 4.28i)21-s + (−3.90 + 6.76i)23-s + (−1.57 − 2.73i)25-s − 17.2i·27-s − 0.0172i·29-s + (−4.79 − 8.31i)31-s + ⋯
L(s)  = 1  + (1.67 − 0.964i)3-s + (0.638 − 1.10i)5-s + (0.515 + 0.857i)7-s + (1.36 − 2.35i)9-s + (−0.332 + 0.191i)11-s + 0.946i·13-s − 2.46i·15-s + (0.477 − 0.275i)17-s + (0.195 + 0.112i)19-s + (1.68 + 0.935i)21-s + (−0.814 + 1.41i)23-s + (−0.315 − 0.546i)25-s − 3.32i·27-s − 0.00320i·29-s + (−0.862 − 1.49i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.376 + 0.926i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.376 + 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $0.376 + 0.926i$
Analytic conductor: \(9.16682\)
Root analytic conductor: \(3.02767\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (737, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1148,\ (\ :1/2),\ 0.376 + 0.926i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.360046569\)
\(L(\frac12)\) \(\approx\) \(3.360046569\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.36 - 2.26i)T \)
41 \( 1 + (-4.71 - 4.32i)T \)
good3 \( 1 + (-2.89 + 1.67i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.42 + 2.47i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.10 - 0.636i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 3.41iT - 13T^{2} \)
17 \( 1 + (-1.97 + 1.13i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.851 - 0.491i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.90 - 6.76i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 0.0172iT - 29T^{2} \)
31 \( 1 + (4.79 + 8.31i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.00 - 6.93i)T + (-18.5 - 32.0i)T^{2} \)
43 \( 1 + 2.35T + 43T^{2} \)
47 \( 1 + (3.58 + 2.06i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.38 - 0.802i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.38 + 5.86i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.36 - 5.82i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (8.17 - 4.71i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 3.13iT - 71T^{2} \)
73 \( 1 + (5.62 + 9.74i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.10 - 0.638i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 3.83T + 83T^{2} \)
89 \( 1 + (-11.9 - 6.87i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 10.5iT - 97T^{2} \)
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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

−9.278598012091262449632072373116, −8.943007252505807016936808696841, −7.971968708097624093940559467876, −7.60018657019341964160699625060, −6.36440936584013669551170405141, −5.43619138461656197208263569925, −4.31607762168075191918327145505, −3.11333258721440900626894942136, −1.95201431666926571483821474294, −1.51518488815893566629465520972, 1.91280802249695537572663294658, 2.89803513419670703986239699748, 3.54109440656487814692132420564, 4.52111503044543064593774712765, 5.56040080875576536183954969632, 6.92867423650086120201066722839, 7.69297065876089074123564839585, 8.311959006569649845088058922863, 9.172416756748423919158048048598, 10.14161824203893458959987778504

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