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Properties

Label 2-1775-1775.1206-c0-0-4
Degree $2$
Conductor $1775$
Sign $0.703 + 0.710i$
Analytic cond. $0.885840$
Root an. cond. $0.941190$
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  + (0.0829 + 0.255i)2-s + (−1.38 + 1.00i)3-s + (0.750 − 0.545i)4-s + (−0.0448 − 0.998i)5-s + (−0.372 − 0.270i)6-s + (0.418 + 0.304i)8-s + (0.601 − 1.85i)9-s + (0.251 − 0.0943i)10-s + (−0.492 + 1.51i)12-s + (1.07 + 1.34i)15-s + (0.243 − 0.750i)16-s + 0.522·18-s + (−1.59 − 1.15i)19-s + (−0.578 − 0.725i)20-s − 0.888·24-s + (−0.995 + 0.0896i)25-s + ⋯
L(s)  = 1  + (0.0829 + 0.255i)2-s + (−1.38 + 1.00i)3-s + (0.750 − 0.545i)4-s + (−0.0448 − 0.998i)5-s + (−0.372 − 0.270i)6-s + (0.418 + 0.304i)8-s + (0.601 − 1.85i)9-s + (0.251 − 0.0943i)10-s + (−0.492 + 1.51i)12-s + (1.07 + 1.34i)15-s + (0.243 − 0.750i)16-s + 0.522·18-s + (−1.59 − 1.15i)19-s + (−0.578 − 0.725i)20-s − 0.888·24-s + (−0.995 + 0.0896i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.703 + 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.703 + 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1775\)    =    \(5^{2} \cdot 71\)
Sign: $0.703 + 0.710i$
Analytic conductor: \(0.885840\)
Root analytic conductor: \(0.941190\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1775} (1206, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1775,\ (\ :0),\ 0.703 + 0.710i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7784422488\)
\(L(\frac12)\) \(\approx\) \(0.7784422488\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.0448 + 0.998i)T \)
71 \( 1 + (0.809 - 0.587i)T \)
good2 \( 1 + (-0.0829 - 0.255i)T + (-0.809 + 0.587i)T^{2} \)
3 \( 1 + (1.38 - 1.00i)T + (0.309 - 0.951i)T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + (0.809 - 0.587i)T^{2} \)
13 \( 1 + (0.809 + 0.587i)T^{2} \)
17 \( 1 + (-0.309 - 0.951i)T^{2} \)
19 \( 1 + (1.59 + 1.15i)T + (0.309 + 0.951i)T^{2} \)
23 \( 1 + (0.809 - 0.587i)T^{2} \)
29 \( 1 + (-0.635 + 0.462i)T + (0.309 - 0.951i)T^{2} \)
31 \( 1 + (-0.309 - 0.951i)T^{2} \)
37 \( 1 + (-0.608 + 1.87i)T + (-0.809 - 0.587i)T^{2} \)
41 \( 1 + (0.809 + 0.587i)T^{2} \)
43 \( 1 + 1.10T + T^{2} \)
47 \( 1 + (-0.309 + 0.951i)T^{2} \)
53 \( 1 + (-0.309 + 0.951i)T^{2} \)
59 \( 1 + (0.809 + 0.587i)T^{2} \)
61 \( 1 + (0.809 - 0.587i)T^{2} \)
67 \( 1 + (-0.309 - 0.951i)T^{2} \)
73 \( 1 + (0.340 + 1.04i)T + (-0.809 + 0.587i)T^{2} \)
79 \( 1 + (-1.55 + 1.13i)T + (0.309 - 0.951i)T^{2} \)
83 \( 1 + (-1.45 - 1.05i)T + (0.309 + 0.951i)T^{2} \)
89 \( 1 + (-0.530 - 1.63i)T + (-0.809 + 0.587i)T^{2} \)
97 \( 1 + (-0.309 + 0.951i)T^{2} \)
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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

−9.477545475679207950983232619474, −8.834053857807381679329305098050, −7.68717254504760283481504756647, −6.59070614982330894155482125911, −6.09555578890039040358508077146, −5.26119051160545906496233630231, −4.72147663404785500933706860082, −3.96084601181161512729872359473, −2.22419695520476626120765727110, −0.67724835863831893016447225533, 1.55363177712371394012751555421, 2.44013549286559649279081019924, 3.58140426308819204841154915035, 4.72697432077877018825033563762, 6.04328275057130128987140896904, 6.37053535204837340304984881493, 7.00305206753984735440479733913, 7.74147461615815442443369162303, 8.422191438526821737823500175197, 10.19204756089726082329712241464

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