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Properties

Label 2-1152-16.11-c2-0-17
Degree $2$
Conductor $1152$
Sign $0.775 - 0.631i$
Analytic cond. $31.3897$
Root an. cond. $5.60265$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.68 + 3.68i)5-s + 9.66·7-s + (−5.51 + 5.51i)11-s + (6.27 − 6.27i)13-s + 6.78·17-s + (13.5 + 13.5i)19-s + 17.0·23-s + 2.17i·25-s + (4.85 − 4.85i)29-s − 5.25i·31-s + (35.6 + 35.6i)35-s + (18.1 + 18.1i)37-s − 48.2i·41-s + (−54.5 + 54.5i)43-s − 40.4i·47-s + ⋯
L(s)  = 1  + (0.737 + 0.737i)5-s + 1.38·7-s + (−0.501 + 0.501i)11-s + (0.482 − 0.482i)13-s + 0.399·17-s + (0.711 + 0.711i)19-s + 0.742·23-s + 0.0868i·25-s + (0.167 − 0.167i)29-s − 0.169i·31-s + (1.01 + 1.01i)35-s + (0.491 + 0.491i)37-s − 1.17i·41-s + (−1.26 + 1.26i)43-s − 0.859i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.775 - 0.631i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.775 - 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.775 - 0.631i$
Analytic conductor: \(31.3897\)
Root analytic conductor: \(5.60265\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (415, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1),\ 0.775 - 0.631i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.787693992\)
\(L(\frac12)\) \(\approx\) \(2.787693992\)
\(L(2)\) 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 \( 1 \)
good5 \( 1 + (-3.68 - 3.68i)T + 25iT^{2} \)
7 \( 1 - 9.66T + 49T^{2} \)
11 \( 1 + (5.51 - 5.51i)T - 121iT^{2} \)
13 \( 1 + (-6.27 + 6.27i)T - 169iT^{2} \)
17 \( 1 - 6.78T + 289T^{2} \)
19 \( 1 + (-13.5 - 13.5i)T + 361iT^{2} \)
23 \( 1 - 17.0T + 529T^{2} \)
29 \( 1 + (-4.85 + 4.85i)T - 841iT^{2} \)
31 \( 1 + 5.25iT - 961T^{2} \)
37 \( 1 + (-18.1 - 18.1i)T + 1.36e3iT^{2} \)
41 \( 1 + 48.2iT - 1.68e3T^{2} \)
43 \( 1 + (54.5 - 54.5i)T - 1.84e3iT^{2} \)
47 \( 1 + 40.4iT - 2.20e3T^{2} \)
53 \( 1 + (-10.8 - 10.8i)T + 2.80e3iT^{2} \)
59 \( 1 + (50.8 - 50.8i)T - 3.48e3iT^{2} \)
61 \( 1 + (-17.0 + 17.0i)T - 3.72e3iT^{2} \)
67 \( 1 + (-22.9 - 22.9i)T + 4.48e3iT^{2} \)
71 \( 1 + 51.6T + 5.04e3T^{2} \)
73 \( 1 + 78.5iT - 5.32e3T^{2} \)
79 \( 1 - 108. iT - 6.24e3T^{2} \)
83 \( 1 + (57.3 + 57.3i)T + 6.88e3iT^{2} \)
89 \( 1 - 44.1iT - 7.92e3T^{2} \)
97 \( 1 - 112.T + 9.40e3T^{2} \)
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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

−9.975549056285867704751760843901, −8.805696236353869293460974284024, −7.954026020574991932568045815153, −7.35941481414978702756157129941, −6.26225530416655322455954904991, −5.42975693425245245983196506232, −4.68385162325867274685260264000, −3.34872415953437987878980189435, −2.28119950917667611394813277304, −1.26281794773343853176502669899, 0.991273404440174421890630557136, 1.85335179355938475298648613854, 3.18314755479663810315066963096, 4.61959622782471950373483026047, 5.15366473511138985739545596505, 5.92817509985066272571417664454, 7.11781398661928696180422860677, 8.026299969423830120230984191337, 8.719580465847036759317022481342, 9.359436359694956641760240870227

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