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Properties

Label 1-4025-4025.328-r0-0-0
Degree $1$
Conductor $4025$
Sign $0.995 - 0.0947i$
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  + (0.996 − 0.0855i)2-s + (0.967 + 0.254i)3-s + (0.985 − 0.170i)4-s + (0.985 + 0.170i)6-s + (0.967 − 0.254i)8-s + (0.870 + 0.491i)9-s + (0.974 − 0.226i)11-s + (0.996 + 0.0855i)12-s + (−0.791 − 0.610i)13-s + (0.941 − 0.336i)16-s + (0.170 − 0.985i)17-s + (0.909 + 0.415i)18-s + (0.897 + 0.441i)19-s + (0.951 − 0.309i)22-s + 24-s + ⋯
L(s)  = 1  + (0.996 − 0.0855i)2-s + (0.967 + 0.254i)3-s + (0.985 − 0.170i)4-s + (0.985 + 0.170i)6-s + (0.967 − 0.254i)8-s + (0.870 + 0.491i)9-s + (0.974 − 0.226i)11-s + (0.996 + 0.0855i)12-s + (−0.791 − 0.610i)13-s + (0.941 − 0.336i)16-s + (0.170 − 0.985i)17-s + (0.909 + 0.415i)18-s + (0.897 + 0.441i)19-s + (0.951 − 0.309i)22-s + 24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4025\)    =    \(5^{2} \cdot 7 \cdot 23\)
Sign: $0.995 - 0.0947i$
Analytic conductor: \(18.6920\)
Root analytic conductor: \(18.6920\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4025} (328, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4025,\ (0:\ ),\ 0.995 - 0.0947i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(5.943519860 - 0.2822090514i\)
\(L(\frac12)\) \(\approx\) \(5.943519860 - 0.2822090514i\)
\(L(1)\) \(\approx\) \(2.973049646 - 0.05563909450i\)
\(L(1)\) \(\approx\) \(2.973049646 - 0.05563909450i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.996 - 0.0855i)T \)
3 \( 1 + (0.967 + 0.254i)T \)
11 \( 1 + (0.974 - 0.226i)T \)
13 \( 1 + (-0.791 - 0.610i)T \)
17 \( 1 + (0.170 - 0.985i)T \)
19 \( 1 + (0.897 + 0.441i)T \)
29 \( 1 + (-0.897 + 0.441i)T \)
31 \( 1 + (0.254 + 0.967i)T \)
37 \( 1 + (-0.491 + 0.870i)T \)
41 \( 1 + (-0.198 - 0.980i)T \)
43 \( 1 + (-0.540 + 0.841i)T \)
47 \( 1 + (-0.951 + 0.309i)T \)
53 \( 1 + (0.999 - 0.0285i)T \)
59 \( 1 + (0.610 - 0.791i)T \)
61 \( 1 + (0.998 + 0.0570i)T \)
67 \( 1 + (0.389 + 0.921i)T \)
71 \( 1 + (0.516 - 0.856i)T \)
73 \( 1 + (-0.717 - 0.696i)T \)
79 \( 1 + (0.564 + 0.825i)T \)
83 \( 1 + (-0.113 - 0.993i)T \)
89 \( 1 + (-0.362 - 0.931i)T \)
97 \( 1 + (-0.113 + 0.993i)T \)
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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

−18.769420252329651027205267082224, −17.710753520086465146017955282785, −16.9513460872028668369386394998, −16.362824029538656069241825857754, −15.40677733907592259475438701777, −14.85018975745709745585644819922, −14.49948475183148865858569155070, −13.69298269762847229244335789293, −13.21170963844671597804801847214, −12.40965222340815986054103156959, −11.85140995845268674839233487213, −11.200488801312744774796050380600, −10.026159248140701858827676969707, −9.56571621776406317516109878609, −8.63791624551524292633644122741, −7.855986496129310151370086240042, −7.11075063039769900458617998917, −6.69226997178203752257194618265, −5.75545026418048580895489482399, −4.86465225667167730847377495015, −3.92051871651784910000472881401, −3.707736573606193575916387822510, −2.57513417459460772656370729739, −1.999219553321778418190319472952, −1.2042890726723681008608821582, 1.131359808681118929346365803583, 1.903751719599398438881414959668, 2.90024764961752723080711184575, 3.32610109225437171089651310529, 4.04783106387856533038953847704, 5.00960698822298250856033708202, 5.38676271742683033453254008861, 6.633705149371182786507312771073, 7.16597252570381919481760629961, 7.85952557455196738509159614925, 8.70349574885005075322237315461, 9.6686878117740011168226812477, 10.0338834112730561623592802737, 10.99470707494592375942155510561, 11.82295954689768518935233568127, 12.32281467577564384038337939889, 13.20998823456865253379647822979, 13.7475733499658104098852853548, 14.55902161224300129606054228270, 14.64603216867218081931741965322, 15.639051193457597264292246760992, 16.176649617185986625935933400744, 16.805885517183145678824230977514, 17.788592885540252918423683592377, 18.75615479477713210704339278203

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