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Properties

Label 2-1472-16.13-c1-0-13
Degree $2$
Conductor $1472$
Sign $-0.650 + 0.759i$
Analytic cond. $11.7539$
Root an. cond. $3.42840$
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  + (−2.38 − 2.38i)3-s + (−1.82 + 1.82i)5-s + 2.41i·7-s + 8.37i·9-s + (−4.14 + 4.14i)11-s + (1.12 + 1.12i)13-s + 8.71·15-s − 1.95·17-s + (−1.61 − 1.61i)19-s + (5.76 − 5.76i)21-s − i·23-s − 1.67i·25-s + (12.8 − 12.8i)27-s + (−4.48 − 4.48i)29-s − 4.66·31-s + ⋯
L(s)  = 1  + (−1.37 − 1.37i)3-s + (−0.817 + 0.817i)5-s + 0.913i·7-s + 2.79i·9-s + (−1.24 + 1.24i)11-s + (0.313 + 0.313i)13-s + 2.24·15-s − 0.475·17-s + (−0.370 − 0.370i)19-s + (1.25 − 1.25i)21-s − 0.208i·23-s − 0.335i·25-s + (2.46 − 2.46i)27-s + (−0.833 − 0.833i)29-s − 0.838·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.650 + 0.759i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.650 + 0.759i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1472\)    =    \(2^{6} \cdot 23\)
Sign: $-0.650 + 0.759i$
Analytic conductor: \(11.7539\)
Root analytic conductor: \(3.42840\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1472} (1105, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1472,\ (\ :1/2),\ -0.650 + 0.759i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.08371560193\)
\(L(\frac12)\) \(\approx\) \(0.08371560193\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
23 \( 1 + iT \)
good3 \( 1 + (2.38 + 2.38i)T + 3iT^{2} \)
5 \( 1 + (1.82 - 1.82i)T - 5iT^{2} \)
7 \( 1 - 2.41iT - 7T^{2} \)
11 \( 1 + (4.14 - 4.14i)T - 11iT^{2} \)
13 \( 1 + (-1.12 - 1.12i)T + 13iT^{2} \)
17 \( 1 + 1.95T + 17T^{2} \)
19 \( 1 + (1.61 + 1.61i)T + 19iT^{2} \)
29 \( 1 + (4.48 + 4.48i)T + 29iT^{2} \)
31 \( 1 + 4.66T + 31T^{2} \)
37 \( 1 + (-4.42 + 4.42i)T - 37iT^{2} \)
41 \( 1 - 3.93iT - 41T^{2} \)
43 \( 1 + (1.65 - 1.65i)T - 43iT^{2} \)
47 \( 1 - 5.02T + 47T^{2} \)
53 \( 1 + (5.60 - 5.60i)T - 53iT^{2} \)
59 \( 1 + (2.15 - 2.15i)T - 59iT^{2} \)
61 \( 1 + (3.69 + 3.69i)T + 61iT^{2} \)
67 \( 1 + (-5.23 - 5.23i)T + 67iT^{2} \)
71 \( 1 + 1.94iT - 71T^{2} \)
73 \( 1 + 8.79iT - 73T^{2} \)
79 \( 1 - 0.555T + 79T^{2} \)
83 \( 1 + (2.11 + 2.11i)T + 83iT^{2} \)
89 \( 1 - 9.22iT - 89T^{2} \)
97 \( 1 - 17.0T + 97T^{2} \)
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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

−9.153010232563323667355607851162, −7.84822772090401808482335091796, −7.58404345802427026385068343888, −6.79134947588014085079284183484, −6.06488636404732300485617164231, −5.26417185025042586705785380250, −4.37728604185744620301319627077, −2.61171670834644337895119431199, −1.94427189439773021816642428216, −0.05896434187712171250393576633, 0.77860626792210101868869407786, 3.39674385171729479759942613094, 3.96964684765214425164335381666, 4.82905373930359089363531344607, 5.45640471164616142270999554530, 6.22164518599770869432767123865, 7.38431157641147399296292010413, 8.358321879313924131492759243898, 9.046877660304638673187159727645, 10.06131678828895738662277289643

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