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Properties

Label 2-1775-1775.1561-c0-0-5
Degree $2$
Conductor $1775$
Sign $-0.359 + 0.933i$
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  + (0.0725 − 0.0527i)2-s + (0.608 − 1.87i)3-s + (−0.306 + 0.943i)4-s + (0.858 − 0.512i)5-s + (−0.0545 − 0.167i)6-s + (0.0552 + 0.169i)8-s + (−2.32 − 1.68i)9-s + (0.0352 − 0.0825i)10-s + (1.57 + 1.14i)12-s + (−0.437 − 1.91i)15-s + (−0.789 − 0.573i)16-s − 0.257·18-s + (−0.340 − 1.04i)19-s + (0.220 + 0.967i)20-s + 0.351·24-s + (0.473 − 0.880i)25-s + ⋯
L(s)  = 1  + (0.0725 − 0.0527i)2-s + (0.608 − 1.87i)3-s + (−0.306 + 0.943i)4-s + (0.858 − 0.512i)5-s + (−0.0545 − 0.167i)6-s + (0.0552 + 0.169i)8-s + (−2.32 − 1.68i)9-s + (0.0352 − 0.0825i)10-s + (1.57 + 1.14i)12-s + (−0.437 − 1.91i)15-s + (−0.789 − 0.573i)16-s − 0.257·18-s + (−0.340 − 1.04i)19-s + (0.220 + 0.967i)20-s + 0.351·24-s + (0.473 − 0.880i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.359 + 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.359 + 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1775\)    =    \(5^{2} \cdot 71\)
Sign: $-0.359 + 0.933i$
Analytic conductor: \(0.885840\)
Root analytic conductor: \(0.941190\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1775} (1561, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1775,\ (\ :0),\ -0.359 + 0.933i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.420004216\)
\(L(\frac12)\) \(\approx\) \(1.420004216\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.858 + 0.512i)T \)
71 \( 1 + (-0.309 + 0.951i)T \)
good2 \( 1 + (-0.0725 + 0.0527i)T + (0.309 - 0.951i)T^{2} \)
3 \( 1 + (-0.608 + 1.87i)T + (-0.809 - 0.587i)T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + (-0.309 + 0.951i)T^{2} \)
13 \( 1 + (-0.309 - 0.951i)T^{2} \)
17 \( 1 + (0.809 - 0.587i)T^{2} \)
19 \( 1 + (0.340 + 1.04i)T + (-0.809 + 0.587i)T^{2} \)
23 \( 1 + (-0.309 + 0.951i)T^{2} \)
29 \( 1 + (-0.0829 + 0.255i)T + (-0.809 - 0.587i)T^{2} \)
31 \( 1 + (0.809 - 0.587i)T^{2} \)
37 \( 1 + (-0.891 - 0.647i)T + (0.309 + 0.951i)T^{2} \)
41 \( 1 + (-0.309 - 0.951i)T^{2} \)
43 \( 1 - 1.50T + T^{2} \)
47 \( 1 + (0.809 + 0.587i)T^{2} \)
53 \( 1 + (0.809 + 0.587i)T^{2} \)
59 \( 1 + (-0.309 - 0.951i)T^{2} \)
61 \( 1 + (-0.309 + 0.951i)T^{2} \)
67 \( 1 + (0.809 - 0.587i)T^{2} \)
73 \( 1 + (1.21 - 0.885i)T + (0.309 - 0.951i)T^{2} \)
79 \( 1 + (0.615 - 1.89i)T + (-0.809 - 0.587i)T^{2} \)
83 \( 1 + (-0.385 - 1.18i)T + (-0.809 + 0.587i)T^{2} \)
89 \( 1 + (1.59 - 1.15i)T + (0.309 - 0.951i)T^{2} \)
97 \( 1 + (0.809 + 0.587i)T^{2} \)
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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

−8.958087191669882617060974436146, −8.399365095512094981433719303164, −7.71129530930934115056202414024, −6.96251128515249153909956654162, −6.28876483281907032872874474065, −5.34907547793380063859524009646, −4.10476499030838587303559212331, −2.76953229640376736805451619407, −2.34444233590609726083045198752, −1.01748479316570184549311030633, 2.00727995763972038392299041344, 3.01072222988566815220612494669, 4.05532752904786869878656417766, 4.71389138282769480991647968676, 5.77728949675770168342634282081, 5.92468133163053533699266793703, 7.44137786074938729128679175514, 8.645755975808404314840636967464, 9.127358293590123239003818197456, 9.840599273862981569516813447131

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