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Properties

Label 2-1848-77.10-c1-0-23
Degree $2$
Conductor $1848$
Sign $0.553 - 0.832i$
Analytic cond. $14.7563$
Root an. cond. $3.84140$
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  + (0.866 + 0.5i)3-s + (2.94 − 1.70i)5-s + (−1.27 + 2.31i)7-s + (0.499 + 0.866i)9-s + (1.01 + 3.15i)11-s − 0.465·13-s + 3.40·15-s + (−1.54 + 2.67i)17-s + (−0.140 − 0.243i)19-s + (−2.26 + 1.37i)21-s + (1.57 + 2.73i)23-s + (3.29 − 5.69i)25-s + 0.999i·27-s + 2.01i·29-s + (7.58 + 4.38i)31-s + ⋯
L(s)  = 1  + (0.499 + 0.288i)3-s + (1.31 − 0.760i)5-s + (−0.481 + 0.876i)7-s + (0.166 + 0.288i)9-s + (0.307 + 0.951i)11-s − 0.129·13-s + 0.878·15-s + (−0.374 + 0.648i)17-s + (−0.0323 − 0.0559i)19-s + (−0.493 + 0.299i)21-s + (0.329 + 0.570i)23-s + (0.658 − 1.13i)25-s + 0.192i·27-s + 0.374i·29-s + (1.36 + 0.786i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.553 - 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1848 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.553 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1848\)    =    \(2^{3} \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.553 - 0.832i$
Analytic conductor: \(14.7563\)
Root analytic conductor: \(3.84140\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1848} (241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1848,\ (\ :1/2),\ 0.553 - 0.832i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.477478088\)
\(L(\frac12)\) \(\approx\) \(2.477478088\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 + (1.27 - 2.31i)T \)
11 \( 1 + (-1.01 - 3.15i)T \)
good5 \( 1 + (-2.94 + 1.70i)T + (2.5 - 4.33i)T^{2} \)
13 \( 1 + 0.465T + 13T^{2} \)
17 \( 1 + (1.54 - 2.67i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.140 + 0.243i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.57 - 2.73i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 2.01iT - 29T^{2} \)
31 \( 1 + (-7.58 - 4.38i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.521 + 0.902i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 6.87T + 41T^{2} \)
43 \( 1 - 4.58iT - 43T^{2} \)
47 \( 1 + (2.73 - 1.58i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.86 + 4.96i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.31 - 0.756i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.45 - 4.25i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.62 + 9.75i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 11.1T + 71T^{2} \)
73 \( 1 + (-2.79 + 4.83i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.83 + 3.36i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 16.3T + 83T^{2} \)
89 \( 1 + (-11.6 + 6.70i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 0.215iT - 97T^{2} \)
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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

−9.301277253202911021942338580984, −8.860098806629702347580226022662, −8.057548623839693766763066560166, −6.80488854049640536862856245127, −6.17504443935271706927205742316, −5.20698517925459703784825595289, −4.66699040913036037841183277216, −3.35827526486182983474763591361, −2.28702485929300978630082709165, −1.55434919617579147974572688281, 0.872057734255058330805277053780, 2.25621830966460032406498262764, 2.99060304547371134869505890121, 3.92734879837498687990487799710, 5.15473163081916009593702085155, 6.31913353608553477976399781189, 6.54494558998547521381628001776, 7.42190709431540963265752614046, 8.402620609390766230514292456114, 9.218656838679845543091231192886

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