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Properties

Label 2-1152-48.35-c1-0-6
Degree $2$
Conductor $1152$
Sign $0.822 + 0.568i$
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  + (−0.236 − 0.236i)5-s − 3.27·7-s + (2.58 − 2.58i)11-s + (1.70 + 1.70i)13-s + 7.05i·17-s + (3.04 − 3.04i)19-s + 1.47i·23-s − 4.88i·25-s + (2.98 − 2.98i)29-s − 8.02i·31-s + (0.774 + 0.774i)35-s + (7.93 − 7.93i)37-s + 2.22·41-s + (−4.61 − 4.61i)43-s + 7.13·47-s + ⋯
L(s)  = 1  + (−0.105 − 0.105i)5-s − 1.23·7-s + (0.778 − 0.778i)11-s + (0.473 + 0.473i)13-s + 1.71i·17-s + (0.697 − 0.697i)19-s + 0.307i·23-s − 0.977i·25-s + (0.554 − 0.554i)29-s − 1.44i·31-s + (0.130 + 0.130i)35-s + (1.30 − 1.30i)37-s + 0.346·41-s + (−0.703 − 0.703i)43-s + 1.04·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.822 + 0.568i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.822 + 0.568i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.822 + 0.568i$
Analytic conductor: \(9.19876\)
Root analytic conductor: \(3.03294\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (863, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1/2),\ 0.822 + 0.568i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.413936189\)
\(L(\frac12)\) \(\approx\) \(1.413936189\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.236 + 0.236i)T + 5iT^{2} \)
7 \( 1 + 3.27T + 7T^{2} \)
11 \( 1 + (-2.58 + 2.58i)T - 11iT^{2} \)
13 \( 1 + (-1.70 - 1.70i)T + 13iT^{2} \)
17 \( 1 - 7.05iT - 17T^{2} \)
19 \( 1 + (-3.04 + 3.04i)T - 19iT^{2} \)
23 \( 1 - 1.47iT - 23T^{2} \)
29 \( 1 + (-2.98 + 2.98i)T - 29iT^{2} \)
31 \( 1 + 8.02iT - 31T^{2} \)
37 \( 1 + (-7.93 + 7.93i)T - 37iT^{2} \)
41 \( 1 - 2.22T + 41T^{2} \)
43 \( 1 + (4.61 + 4.61i)T + 43iT^{2} \)
47 \( 1 - 7.13T + 47T^{2} \)
53 \( 1 + (-5.81 - 5.81i)T + 53iT^{2} \)
59 \( 1 + (-7.46 + 7.46i)T - 59iT^{2} \)
61 \( 1 + (-4.04 - 4.04i)T + 61iT^{2} \)
67 \( 1 + (2.90 - 2.90i)T - 67iT^{2} \)
71 \( 1 - 1.02iT - 71T^{2} \)
73 \( 1 - 4.08iT - 73T^{2} \)
79 \( 1 + 5.36iT - 79T^{2} \)
83 \( 1 + (3.93 + 3.93i)T + 83iT^{2} \)
89 \( 1 + 2.35T + 89T^{2} \)
97 \( 1 - 9.97T + 97T^{2} \)
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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

−9.638396176916323816418999082488, −8.959183056806501026579957958018, −8.218060722460752803725933092051, −7.12436940382630344270450585030, −6.17330506269534942319657797257, −5.87405016636290128594211374098, −4.17586591098028960389821158346, −3.67884706536597731080136646602, −2.42665702624789474087059212840, −0.77339736739857331467886736487, 1.10583423827330864793672717996, 2.83463289698169315907249786812, 3.50980565990878182140485199887, 4.70797754461585339773309369184, 5.68490856103856302730047268259, 6.76063917732106064308933671876, 7.12243130298792404511121494064, 8.272812025740919230284182910045, 9.325923531847800494879758852041, 9.707248719932353231821847238175

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