Properties

Label 2-2352-21.11-c0-0-0
Degree $2$
Conductor $2352$
Sign $0.386 - 0.922i$
Analytic cond. $1.17380$
Root an. cond. $1.08342$
Motivic weight $0$
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 + (−0.499 − 0.866i)9-s + 13-s + (0.5 + 0.866i)19-s + (−0.5 + 0.866i)25-s + 0.999·27-s + (0.5 − 0.866i)31-s + (0.5 + 0.866i)37-s + (−0.5 + 0.866i)39-s + 43-s − 0.999·57-s + (1 + 1.73i)61-s + (−0.5 + 0.866i)67-s + (−0.5 + 0.866i)73-s + (−0.499 − 0.866i)75-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)3-s + (−0.499 − 0.866i)9-s + 13-s + (0.5 + 0.866i)19-s + (−0.5 + 0.866i)25-s + 0.999·27-s + (0.5 − 0.866i)31-s + (0.5 + 0.866i)37-s + (−0.5 + 0.866i)39-s + 43-s − 0.999·57-s + (1 + 1.73i)61-s + (−0.5 + 0.866i)67-s + (−0.5 + 0.866i)73-s + (−0.499 − 0.866i)75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $0.386 - 0.922i$
Analytic conductor: \(1.17380\)
Root analytic conductor: \(1.08342\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2352} (2321, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2352,\ (\ :0),\ 0.386 - 0.922i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.014151757\)
\(L(\frac12)\) \(\approx\) \(1.014151757\)
\(L(1)\) 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 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 - T + T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T + T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + 2T + T^{2} \)
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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

−9.426353997519676712811974328904, −8.653870367119313653659351225660, −7.890531584242956379991945851906, −6.89180192732979288392052849883, −5.89148518645994271521415927600, −5.58931091868301816757841368845, −4.39180957848522038347555365599, −3.80398371324305062295748656594, −2.84695028704436942860821750156, −1.25098469745794540983897487373, 0.875647314138879257351350989160, 2.07474686449359264324201125260, 3.10690787491579656064697987943, 4.29217243239320374544618641719, 5.23411465459115133808916164584, 6.03500552218341294440449961167, 6.65035950186296513439070104807, 7.44378509326576439303765332301, 8.186813555588697684134805712111, 8.861183849603686739821246294801

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