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Properties

Label 1-4025-4025.1623-r0-0-0
Degree $1$
Conductor $4025$
Sign $-0.658 - 0.752i$
Analytic cond. $18.6920$
Root an. cond. $18.6920$
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  + (0.441 − 0.897i)2-s + (−0.980 − 0.198i)3-s + (−0.610 − 0.791i)4-s + (−0.610 + 0.791i)6-s + (−0.980 + 0.198i)8-s + (0.921 + 0.389i)9-s + (−0.985 − 0.170i)11-s + (0.441 + 0.897i)12-s + (0.633 − 0.774i)13-s + (−0.254 + 0.967i)16-s + (0.791 + 0.610i)17-s + (0.755 − 0.654i)18-s + (0.941 − 0.336i)19-s + (−0.587 + 0.809i)22-s + 24-s + ⋯
L(s)  = 1  + (0.441 − 0.897i)2-s + (−0.980 − 0.198i)3-s + (−0.610 − 0.791i)4-s + (−0.610 + 0.791i)6-s + (−0.980 + 0.198i)8-s + (0.921 + 0.389i)9-s + (−0.985 − 0.170i)11-s + (0.441 + 0.897i)12-s + (0.633 − 0.774i)13-s + (−0.254 + 0.967i)16-s + (0.791 + 0.610i)17-s + (0.755 − 0.654i)18-s + (0.941 − 0.336i)19-s + (−0.587 + 0.809i)22-s + 24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.658 - 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.658 - 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4025\)    =    \(5^{2} \cdot 7 \cdot 23\)
Sign: $-0.658 - 0.752i$
Analytic conductor: \(18.6920\)
Root analytic conductor: \(18.6920\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4025} (1623, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4025,\ (0:\ ),\ -0.658 - 0.752i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5066232802 - 1.117263837i\)
\(L(\frac12)\) \(\approx\) \(0.5066232802 - 1.117263837i\)
\(L(1)\) \(\approx\) \(0.7183513640 - 0.5439616644i\)
\(L(1)\) \(\approx\) \(0.7183513640 - 0.5439616644i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.441 - 0.897i)T \)
3 \( 1 + (-0.980 - 0.198i)T \)
11 \( 1 + (-0.985 - 0.170i)T \)
13 \( 1 + (0.633 - 0.774i)T \)
17 \( 1 + (0.791 + 0.610i)T \)
19 \( 1 + (0.941 - 0.336i)T \)
29 \( 1 + (-0.941 - 0.336i)T \)
31 \( 1 + (-0.198 - 0.980i)T \)
37 \( 1 + (-0.389 + 0.921i)T \)
41 \( 1 + (-0.516 + 0.856i)T \)
43 \( 1 + (-0.909 + 0.415i)T \)
47 \( 1 + (0.587 - 0.809i)T \)
53 \( 1 + (0.931 - 0.362i)T \)
59 \( 1 + (0.774 + 0.633i)T \)
61 \( 1 + (0.736 + 0.676i)T \)
67 \( 1 + (-0.884 + 0.466i)T \)
71 \( 1 + (0.696 - 0.717i)T \)
73 \( 1 + (0.825 + 0.564i)T \)
79 \( 1 + (0.998 + 0.0570i)T \)
83 \( 1 + (-0.996 - 0.0855i)T \)
89 \( 1 + (0.993 - 0.113i)T \)
97 \( 1 + (-0.996 + 0.0855i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.3830386768787829100670510805, −18.04609418187072181336234900168, −17.16639172426535662592725274193, −16.53541514740764877045679257300, −15.98264852602853903780648336260, −15.647740725827714062679574163366, −14.69777527183329627883215201805, −13.956848840730971368482418482333, −13.35801062613052782412741125818, −12.49666786197353205209978900068, −12.06168447463171841510896918090, −11.26942769425124640136590904768, −10.48200343166470437399637340828, −9.65541290600356589655534205268, −9.013824417510753787550099059303, −8.0393542038599167393441761457, −7.23707351816865822077121145651, −6.87046320956739088677080626550, −5.813046832439173640507822715555, −5.369169613907523007526906376401, −4.84748202690931473364980582140, −3.820043911362477655856692773680, −3.312914963299278771050112002137, −1.96653872674107040871086318272, −0.7406828774714133921246996497, 0.522254842163306343766056047580, 1.27822876895565879721851476084, 2.19587929581551657749456101990, 3.17314855409202668728325397301, 3.83597959991438086284248593668, 4.81903766227579574135088427581, 5.56941521275045336865199479966, 5.75465975397633752448482641687, 6.8187191866637934682416122010, 7.79367115547424065732923715376, 8.436941644155332773255077984131, 9.640397107916587128463306398845, 10.179291523184996312425797496675, 10.69693354599434286536123043520, 11.50323612653035939422002487749, 11.86540248138420261348949178934, 12.831752700193196908678307541780, 13.24110031131292132792317065469, 13.69866608277578620468905873888, 14.99202373758832349862700714299, 15.31160171196942609180261706870, 16.2547336937097581938750416400, 16.92128849463468545124047715023, 17.81840953688478305842443011088, 18.38285043683940407970551544870

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