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Properties

Label 2-1872-52.3-c0-0-1
Degree $2$
Conductor $1872$
Sign $0.964 - 0.265i$
Analytic cond. $0.934249$
Root an. cond. $0.966565$
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  + (1.5 + 0.866i)7-s + (0.5 − 0.866i)13-s − 25-s + 1.73i·31-s + (−1 − 1.73i)37-s + (1.5 + 0.866i)43-s + (1 + 1.73i)49-s + (0.5 − 0.866i)61-s + (−1.5 + 0.866i)67-s + 73-s − 1.73i·79-s + (1.5 − 0.866i)91-s + (−0.5 + 0.866i)97-s − 1.73i·103-s + 109-s + ⋯
L(s)  = 1  + (1.5 + 0.866i)7-s + (0.5 − 0.866i)13-s − 25-s + 1.73i·31-s + (−1 − 1.73i)37-s + (1.5 + 0.866i)43-s + (1 + 1.73i)49-s + (0.5 − 0.866i)61-s + (−1.5 + 0.866i)67-s + 73-s − 1.73i·79-s + (1.5 − 0.866i)91-s + (−0.5 + 0.866i)97-s − 1.73i·103-s + 109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1872\)    =    \(2^{4} \cdot 3^{2} \cdot 13\)
Sign: $0.964 - 0.265i$
Analytic conductor: \(0.934249\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1872} (991, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1872,\ (\ :0),\ 0.964 - 0.265i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.363923635\)
\(L(\frac12)\) \(\approx\) \(1.363923635\)
\(L(1)\) 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 \( 1 \)
13 \( 1 + (-0.5 + 0.866i)T \)
good5 \( 1 + T^{2} \)
7 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T^{2} \)
31 \( 1 - 1.73iT - T^{2} \)
37 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 - T + T^{2} \)
79 \( 1 + 1.73iT - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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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

−9.224149584016675446360213320961, −8.644571181022881467614383252888, −7.960565888230562308476823668418, −7.30836543095103973633524438045, −6.03679011790647472999521587013, −5.44103727780616589296323973804, −4.69896039456550899348930373970, −3.60439194241171661245682692623, −2.43507767797849144951779297498, −1.44441558691951241395809668698, 1.28726969413034346672006975929, 2.24104959990136408578147569712, 3.82436215351258823651405071618, 4.35232971902048137169990116015, 5.25954271343067211816876073378, 6.22187206474241872117531651773, 7.17997492775918182473730595135, 7.83536660761551778622004138928, 8.478207623685045489361784918884, 9.356309588090760038477551705427

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