Properties

Label 1-4025-4025.202-r0-0-0
Degree $1$
Conductor $4025$
Sign $0.999 + 0.00593i$
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  + (0.633 + 0.774i)2-s + (−0.884 − 0.466i)3-s + (−0.198 + 0.980i)4-s + (−0.198 − 0.980i)6-s + (−0.884 + 0.466i)8-s + (0.564 + 0.825i)9-s + (−0.254 − 0.967i)11-s + (0.633 − 0.774i)12-s + (0.856 + 0.516i)13-s + (−0.921 − 0.389i)16-s + (0.980 − 0.198i)17-s + (−0.281 + 0.959i)18-s + (−0.870 − 0.491i)19-s + (0.587 − 0.809i)22-s + 24-s + ⋯
L(s)  = 1  + (0.633 + 0.774i)2-s + (−0.884 − 0.466i)3-s + (−0.198 + 0.980i)4-s + (−0.198 − 0.980i)6-s + (−0.884 + 0.466i)8-s + (0.564 + 0.825i)9-s + (−0.254 − 0.967i)11-s + (0.633 − 0.774i)12-s + (0.856 + 0.516i)13-s + (−0.921 − 0.389i)16-s + (0.980 − 0.198i)17-s + (−0.281 + 0.959i)18-s + (−0.870 − 0.491i)19-s + (0.587 − 0.809i)22-s + 24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00593i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00593i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4025\)    =    \(5^{2} \cdot 7 \cdot 23\)
Sign: $0.999 + 0.00593i$
Analytic conductor: \(18.6920\)
Root analytic conductor: \(18.6920\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4025} (202, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4025,\ (0:\ ),\ 0.999 + 0.00593i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.521877852 + 0.004516032140i\)
\(L(\frac12)\) \(\approx\) \(1.521877852 + 0.004516032140i\)
\(L(1)\) \(\approx\) \(1.045941478 + 0.2786005551i\)
\(L(1)\) \(\approx\) \(1.045941478 + 0.2786005551i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.633 + 0.774i)T \)
3 \( 1 + (-0.884 - 0.466i)T \)
11 \( 1 + (-0.254 - 0.967i)T \)
13 \( 1 + (0.856 + 0.516i)T \)
17 \( 1 + (0.980 - 0.198i)T \)
19 \( 1 + (-0.870 - 0.491i)T \)
29 \( 1 + (0.870 - 0.491i)T \)
31 \( 1 + (0.466 + 0.884i)T \)
37 \( 1 + (0.825 - 0.564i)T \)
41 \( 1 + (0.0285 + 0.999i)T \)
43 \( 1 + (-0.989 + 0.142i)T \)
47 \( 1 + (-0.587 + 0.809i)T \)
53 \( 1 + (-0.226 - 0.974i)T \)
59 \( 1 + (0.516 - 0.856i)T \)
61 \( 1 + (-0.897 - 0.441i)T \)
67 \( 1 + (-0.0570 - 0.998i)T \)
71 \( 1 + (-0.362 - 0.931i)T \)
73 \( 1 + (0.113 + 0.993i)T \)
79 \( 1 + (-0.0855 - 0.996i)T \)
83 \( 1 + (0.791 + 0.610i)T \)
89 \( 1 + (-0.985 - 0.170i)T \)
97 \( 1 + (0.791 - 0.610i)T \)
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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

−18.4670269663241742199485790272, −17.931898734969381629304675724634, −17.144942331220903110362631362594, −16.40399733733370316044179593570, −15.56919321368619866884947771272, −15.06666581231687994870630591953, −14.50310540973859824737276351131, −13.42528782120467984511798592981, −12.89093356082898539052605603809, −12.150381210152165527967604980578, −11.78096225891562624139316190586, −10.82867954509284843357295982964, −10.28512198747806926342686743817, −9.95187265488032006036636384181, −8.99794969125943435801085748629, −8.07574367949310046346251089008, −6.96960305387789320471274431430, −6.18553801981233472361375677816, −5.65271990224088772914338920835, −4.89231805282181582287775453215, −4.21380686963742764746731659041, −3.58769376862204047954045658808, −2.67349743224017005449090785530, −1.60738169408950905182061163698, −0.872908038965238038519533593402, 0.49325183854232675193926579083, 1.591086531324729732957606768197, 2.77360710823431676217288750755, 3.52639474569378236705968620564, 4.54607869546556307020026478125, 5.03657355963547480124763453384, 6.03273026032966869767819129383, 6.29108367229505267708653290512, 6.99356721111614886691990078284, 8.045478322839012487045245022117, 8.28183385960920723736281698552, 9.34007803154864499335351873234, 10.360944780460388009534850658335, 11.25361521724411286456430156164, 11.58489874287920941749375790185, 12.50279960302127064540589395069, 13.02389063820672428674882323529, 13.73526705381500699897102720982, 14.19858154959753923822934544703, 15.164508318689397733053628695, 16.00429872421800680290767273743, 16.36195799606015631257772900158, 16.91481140513217091623871962343, 17.73629333395998474091841111209, 18.2527960872570631953782546188

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