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Properties

Label 2-1872-156.119-c0-0-1
Degree $2$
Conductor $1872$
Sign $0.556 + 0.831i$
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  + (−0.707 − 0.707i)5-s + (0.5 + 0.866i)13-s + (0.965 − 1.67i)17-s + (−0.448 + 0.258i)29-s + (0.5 − 1.86i)37-s + (0.448 − 1.67i)41-s + (0.866 − 0.5i)49-s + 0.517i·53-s + (−0.5 + 0.866i)61-s + (0.258 − 0.965i)65-s + (−0.366 − 0.366i)73-s + (−1.86 + 0.5i)85-s + (−1.93 − 0.517i)89-s + (0.366 + 1.36i)97-s + (0.965 + 1.67i)101-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)5-s + (0.5 + 0.866i)13-s + (0.965 − 1.67i)17-s + (−0.448 + 0.258i)29-s + (0.5 − 1.86i)37-s + (0.448 − 1.67i)41-s + (0.866 − 0.5i)49-s + 0.517i·53-s + (−0.5 + 0.866i)61-s + (0.258 − 0.965i)65-s + (−0.366 − 0.366i)73-s + (−1.86 + 0.5i)85-s + (−1.93 − 0.517i)89-s + (0.366 + 1.36i)97-s + (0.965 + 1.67i)101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.556 + 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.556 + 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1872\)    =    \(2^{4} \cdot 3^{2} \cdot 13\)
Sign: $0.556 + 0.831i$
Analytic conductor: \(0.934249\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1872} (431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1872,\ (\ :0),\ 0.556 + 0.831i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.003277967\)
\(L(\frac12)\) \(\approx\) \(1.003277967\)
\(L(1)\) 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 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
7 \( 1 + (-0.866 + 0.5i)T^{2} \)
11 \( 1 + (-0.866 - 0.5i)T^{2} \)
17 \( 1 + (-0.965 + 1.67i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.866 - 0.5i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.448 - 0.258i)T + (0.5 - 0.866i)T^{2} \)
31 \( 1 + iT^{2} \)
37 \( 1 + (-0.5 + 1.86i)T + (-0.866 - 0.5i)T^{2} \)
41 \( 1 + (-0.448 + 1.67i)T + (-0.866 - 0.5i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 - 0.517iT - T^{2} \)
59 \( 1 + (0.866 - 0.5i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.866 - 0.5i)T^{2} \)
71 \( 1 + (-0.866 + 0.5i)T^{2} \)
73 \( 1 + (0.366 + 0.366i)T + iT^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + (1.93 + 0.517i)T + (0.866 + 0.5i)T^{2} \)
97 \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \)
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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

−9.061061646021698896524688568300, −8.746379215077930883877215070430, −7.51605458738415921466398153742, −7.28312919271142887458709880223, −5.99912894979101797793164851256, −5.20287581705692144380683655781, −4.31816469667403411058172394207, −3.58655559361355808467770415431, −2.31948397193740713601295863855, −0.843146133039978458504499634867, 1.40845621313387123612021937459, 2.96798788247115091037400467971, 3.56116110526454750872983550992, 4.50548970662966015115204524158, 5.72739844306191333795445746318, 6.28909154567699655060071924809, 7.31365724711466279676719255599, 8.036036610785999870763598567055, 8.459076740171270628509899436987, 9.741239634285486443429458139397

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