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Properties

Label 2-2352-7.4-c1-0-33
Degree $2$
Conductor $2352$
Sign $-0.701 + 0.712i$
Analytic cond. $18.7808$
Root an. cond. $4.33368$
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  + (0.5 − 0.866i)3-s + (1 + 1.73i)5-s + (−0.499 − 0.866i)9-s − 6·13-s + 1.99·15-s + (−1 + 1.73i)17-s + (−2 − 3.46i)19-s + (−2 − 3.46i)23-s + (0.500 − 0.866i)25-s − 0.999·27-s − 10·29-s + (4 − 6.92i)31-s + (−3 − 5.19i)37-s + (−3 + 5.19i)39-s + 2·41-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + (0.447 + 0.774i)5-s + (−0.166 − 0.288i)9-s − 1.66·13-s + 0.516·15-s + (−0.242 + 0.420i)17-s + (−0.458 − 0.794i)19-s + (−0.417 − 0.722i)23-s + (0.100 − 0.173i)25-s − 0.192·27-s − 1.85·29-s + (0.718 − 1.24i)31-s + (−0.493 − 0.854i)37-s + (−0.480 + 0.832i)39-s + 0.312·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $-0.701 + 0.712i$
Analytic conductor: \(18.7808\)
Root analytic conductor: \(4.33368\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2352} (1537, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2352,\ (\ :1/2),\ -0.701 + 0.712i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8774480406\)
\(L(\frac12)\) \(\approx\) \(0.8774480406\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 \)
good5 \( 1 + (-1 - 1.73i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 6T + 13T^{2} \)
17 \( 1 + (1 - 1.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2 + 3.46i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2 + 3.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 10T + 29T^{2} \)
31 \( 1 + (-4 + 6.92i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3 + 5.19i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + (4 + 6.92i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5 + 8.66i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (6 - 10.3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1 + 1.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6 + 10.3i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + (7 - 12.1i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 12T + 83T^{2} \)
89 \( 1 + (1 + 1.73i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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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

−8.696968124891315927604816933819, −7.76506429419822287947220711457, −7.15689287777917210912354366575, −6.50963129853844191391204897558, −5.70692759499162746103138787486, −4.69950536536891106020266045636, −3.70230190531227872927514265696, −2.38679290427768111827473991697, −2.22989096279616968007982541764, −0.25954707695368090163536175168, 1.53735686046447107172648826928, 2.53675695643368377915193173712, 3.61131101873430954589346341482, 4.64847948427257203652237315370, 5.16159610909159105572984952089, 5.96627383423334745831303693689, 7.11228231587564261040551580900, 7.78996411351314753937867133967, 8.625662616132641324420303275209, 9.472349341604462403841358490056

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