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Properties

Label 2-1152-96.35-c1-0-15
Degree $2$
Conductor $1152$
Sign $-0.966 + 0.258i$
Analytic cond. $9.19876$
Root an. cond. $3.03294$
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  + (−0.0505 + 0.0209i)5-s + (−1.44 − 1.44i)7-s + (−0.320 + 0.132i)11-s + (0.623 − 1.50i)13-s − 5.65·17-s + (−4.13 − 1.71i)19-s + (3.03 + 3.03i)23-s + (−3.53 + 3.53i)25-s + (−0.721 + 1.74i)29-s − 5.26i·31-s + (0.103 + 0.0427i)35-s + (−1.32 − 3.18i)37-s + (−6.90 + 6.90i)41-s + (−3.40 − 8.21i)43-s + 3.23i·47-s + ⋯
L(s)  = 1  + (−0.0226 + 0.00936i)5-s + (−0.544 − 0.544i)7-s + (−0.0967 + 0.0400i)11-s + (0.172 − 0.417i)13-s − 1.37·17-s + (−0.947 − 0.392i)19-s + (0.632 + 0.632i)23-s + (−0.706 + 0.706i)25-s + (−0.134 + 0.323i)29-s − 0.945i·31-s + (0.0174 + 0.00721i)35-s + (−0.217 − 0.524i)37-s + (−1.07 + 1.07i)41-s + (−0.518 − 1.25i)43-s + 0.471i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.966 + 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.966 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $-0.966 + 0.258i$
Analytic conductor: \(9.19876\)
Root analytic conductor: \(3.03294\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1/2),\ -0.966 + 0.258i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3209885123\)
\(L(\frac12)\) \(\approx\) \(0.3209885123\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (0.0505 - 0.0209i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (1.44 + 1.44i)T + 7iT^{2} \)
11 \( 1 + (0.320 - 0.132i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + (-0.623 + 1.50i)T + (-9.19 - 9.19i)T^{2} \)
17 \( 1 + 5.65T + 17T^{2} \)
19 \( 1 + (4.13 + 1.71i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (-3.03 - 3.03i)T + 23iT^{2} \)
29 \( 1 + (0.721 - 1.74i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + 5.26iT - 31T^{2} \)
37 \( 1 + (1.32 + 3.18i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (6.90 - 6.90i)T - 41iT^{2} \)
43 \( 1 + (3.40 + 8.21i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 - 3.23iT - 47T^{2} \)
53 \( 1 + (-0.579 - 1.40i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (4.21 + 10.1i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (12.1 + 5.03i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + (3.34 - 8.08i)T + (-47.3 - 47.3i)T^{2} \)
71 \( 1 + (9.36 - 9.36i)T - 71iT^{2} \)
73 \( 1 + (1.72 + 1.72i)T + 73iT^{2} \)
79 \( 1 - 15.1T + 79T^{2} \)
83 \( 1 + (-2.48 + 6.00i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (-2.70 - 2.70i)T + 89iT^{2} \)
97 \( 1 - 6.43T + 97T^{2} \)
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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

−9.384104921558407148844695853138, −8.709430932747403359168958326670, −7.70386111600374483556056868837, −6.90068988930179495342765156925, −6.18033108812575744557971881469, −5.08232022604076396092770807547, −4.10420486091701895484695966660, −3.18141773780554555125344008771, −1.88958659022656928135324116766, −0.12907991080769879845465196931, 1.89186219477967855442486423977, 2.92946611776493764311267182770, 4.12779241483167426506112219646, 4.97001661445264855623194984673, 6.27286692591900474090001224127, 6.55052404429326705214137377956, 7.77086084922458581869704330943, 8.778031137057043596556289354145, 9.103806479094477531927606460547, 10.28805758410996847307765040859

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