Properties

Degree $2$
Conductor $2352$
Sign $-0.900 - 0.435i$
Motivic weight $1$
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 + (−1.70 − 2.95i)5-s + (−0.499 − 0.866i)9-s + (−1 + 1.73i)11-s + 2.58·13-s − 3.41·15-s + (1.12 − 1.94i)17-s + (−1.41 − 2.44i)19-s + (−3.82 − 6.63i)23-s + (−3.32 + 5.76i)25-s − 0.999·27-s − 6.82·29-s + (−0.585 + 1.01i)31-s + (0.999 + 1.73i)33-s + (2 + 3.46i)37-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + (−0.763 − 1.32i)5-s + (−0.166 − 0.288i)9-s + (−0.301 + 0.522i)11-s + 0.717·13-s − 0.881·15-s + (0.271 − 0.471i)17-s + (−0.324 − 0.561i)19-s + (−0.798 − 1.38i)23-s + (−0.665 + 1.15i)25-s − 0.192·27-s − 1.26·29-s + (−0.105 + 0.182i)31-s + (0.174 + 0.301i)33-s + (0.328 + 0.569i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.900 - 0.435i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.900 - 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $-0.900 - 0.435i$
Motivic weight: \(1\)
Character: $\chi_{2352} (1537, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2352,\ (\ :1/2),\ -0.900 - 0.435i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6834253136\)
\(L(\frac12)\) \(\approx\) \(0.6834253136\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.70 + 2.95i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 2.58T + 13T^{2} \)
17 \( 1 + (-1.12 + 1.94i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.41 + 2.44i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.82 + 6.63i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 6.82T + 29T^{2} \)
31 \( 1 + (0.585 - 1.01i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2 - 3.46i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 6.24T + 41T^{2} \)
43 \( 1 + 5.65T + 43T^{2} \)
47 \( 1 + (1.41 + 2.44i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1 + 1.73i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.585 - 1.01i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.12 + 10.6i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.82 - 4.89i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 9.31T + 71T^{2} \)
73 \( 1 + (6.94 - 12.0i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.82 - 11.8i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 7.31T + 83T^{2} \)
89 \( 1 + (-7.12 - 12.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 2.58T + 97T^{2} \)
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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

−8.449219698459344939756476535867, −7.948579431692616415563814484343, −7.16653593212079323796056702970, −6.25501695047126728192532491829, −5.26464166278851014294869468191, −4.49222365447608573408195898954, −3.76603456855653435810253070265, −2.51973511172944481287662037351, −1.36358588209344837365117866681, −0.22519150083049940317497307569, 1.84515130578861641669598011811, 3.13446574965654026121654987078, 3.57813896172086589162832373592, 4.33370124531380089695385918679, 5.77608615893460746556427438818, 6.12828046890969184957813607322, 7.47509567713771011945610394565, 7.67301407763802092815499323230, 8.609248533025890810441998619930, 9.437566814394664292374963045447

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