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Properties

Label 1-4025-4025.748-r0-0-0
Degree $1$
Conductor $4025$
Sign $0.716 - 0.697i$
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  + (−0.676 + 0.736i)2-s + (0.791 − 0.610i)3-s + (−0.0855 − 0.996i)4-s + (−0.0855 + 0.996i)6-s + (0.791 + 0.610i)8-s + (0.254 − 0.967i)9-s + (0.993 − 0.113i)11-s + (−0.676 − 0.736i)12-s + (−0.441 − 0.897i)13-s + (−0.985 + 0.170i)16-s + (0.996 + 0.0855i)17-s + (0.540 + 0.841i)18-s + (0.974 + 0.226i)19-s + (−0.587 + 0.809i)22-s + 24-s + ⋯
L(s)  = 1  + (−0.676 + 0.736i)2-s + (0.791 − 0.610i)3-s + (−0.0855 − 0.996i)4-s + (−0.0855 + 0.996i)6-s + (0.791 + 0.610i)8-s + (0.254 − 0.967i)9-s + (0.993 − 0.113i)11-s + (−0.676 − 0.736i)12-s + (−0.441 − 0.897i)13-s + (−0.985 + 0.170i)16-s + (0.996 + 0.0855i)17-s + (0.540 + 0.841i)18-s + (0.974 + 0.226i)19-s + (−0.587 + 0.809i)22-s + 24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.716 - 0.697i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.716 - 0.697i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4025\)    =    \(5^{2} \cdot 7 \cdot 23\)
Sign: $0.716 - 0.697i$
Analytic conductor: \(18.6920\)
Root analytic conductor: \(18.6920\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4025} (748, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4025,\ (0:\ ),\ 0.716 - 0.697i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.728094013 - 0.7018903929i\)
\(L(\frac12)\) \(\approx\) \(1.728094013 - 0.7018903929i\)
\(L(1)\) \(\approx\) \(1.117702632 - 0.07722484311i\)
\(L(1)\) \(\approx\) \(1.117702632 - 0.07722484311i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.676 + 0.736i)T \)
3 \( 1 + (0.791 - 0.610i)T \)
11 \( 1 + (0.993 - 0.113i)T \)
13 \( 1 + (-0.441 - 0.897i)T \)
17 \( 1 + (0.996 + 0.0855i)T \)
19 \( 1 + (0.974 + 0.226i)T \)
29 \( 1 + (-0.974 + 0.226i)T \)
31 \( 1 + (-0.610 + 0.791i)T \)
37 \( 1 + (0.967 + 0.254i)T \)
41 \( 1 + (-0.774 - 0.633i)T \)
43 \( 1 + (0.281 - 0.959i)T \)
47 \( 1 + (0.587 - 0.809i)T \)
53 \( 1 + (0.717 + 0.696i)T \)
59 \( 1 + (0.897 - 0.441i)T \)
61 \( 1 + (0.0285 - 0.999i)T \)
67 \( 1 + (0.980 - 0.198i)T \)
71 \( 1 + (-0.870 + 0.491i)T \)
73 \( 1 + (0.389 + 0.921i)T \)
79 \( 1 + (0.466 - 0.884i)T \)
83 \( 1 + (0.0570 + 0.998i)T \)
89 \( 1 + (-0.564 + 0.825i)T \)
97 \( 1 + (0.0570 - 0.998i)T \)
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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

−18.78904235245649194186607268598, −18.03994199754599322963964873500, −17.11236150563034599686114230452, −16.46011935320163828023359563257, −16.234307705174501278205822224620, −14.97576380683531097409199626310, −14.5464743655405142450179505850, −13.73782846952087107749575874672, −13.12572921440436474015979879507, −12.18160710622414392136537995371, −11.54654082721190829791850212969, −11.016554374100984223049726363456, −9.9242219682547401667624011097, −9.5663028708631094705258767326, −9.149737505748369990905952996077, −8.27871513372518847709796080808, −7.522590274995643856782349466472, −7.04879730773104291349270985191, −5.78716557136068511737307041398, −4.70587664429288211411844338843, −4.043607059167645025253211869348, −3.44549696813768533468535000351, −2.627233794955655433904768559982, −1.853680364849745240093025266685, −1.04609534642629538269742906713, 0.69846335183631689392888000825, 1.37266685913645022673850298484, 2.23337723341425290610047044313, 3.2970849574351998427664336874, 3.95377232645650890084509638576, 5.30929115578189467405818817747, 5.71523821772177356863065006220, 6.787008460723809186011329778165, 7.23794719214494273309497882096, 7.90651094677056490417640641600, 8.53317593871736732097310355775, 9.285455881340584274148674904350, 9.787444003690632625010260261, 10.5354492852556252714046304824, 11.58205066971194701134828805558, 12.2670382032113948824175646276, 13.044741516455062796158221251152, 13.97823288857764353268024657590, 14.29740941175448487602229551587, 15.016585863800507455920910483727, 15.52956112598731131587675075389, 16.513788842591873745054335829597, 17.03931967203686629940167803392, 17.76463562747805334945576748065, 18.45259629530231044180108947415

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