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Properties

Label 1-4025-4025.842-r0-0-0
Degree $1$
Conductor $4025$
Sign $0.937 + 0.348i$
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  + (0.151 + 0.988i)2-s + (0.556 + 0.830i)3-s + (−0.953 + 0.299i)4-s + (−0.736 + 0.676i)6-s + (−0.441 − 0.897i)8-s + (−0.380 + 0.924i)9-s + (−0.449 + 0.893i)11-s + (−0.780 − 0.625i)12-s + (−0.226 − 0.974i)13-s + (0.820 − 0.572i)16-s + (−0.976 + 0.217i)17-s + (−0.971 − 0.235i)18-s + (−0.398 − 0.917i)19-s + (−0.951 − 0.309i)22-s + (0.5 − 0.866i)24-s + ⋯
L(s)  = 1  + (0.151 + 0.988i)2-s + (0.556 + 0.830i)3-s + (−0.953 + 0.299i)4-s + (−0.736 + 0.676i)6-s + (−0.441 − 0.897i)8-s + (−0.380 + 0.924i)9-s + (−0.449 + 0.893i)11-s + (−0.780 − 0.625i)12-s + (−0.226 − 0.974i)13-s + (0.820 − 0.572i)16-s + (−0.976 + 0.217i)17-s + (−0.971 − 0.235i)18-s + (−0.398 − 0.917i)19-s + (−0.951 − 0.309i)22-s + (0.5 − 0.866i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.937 + 0.348i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.937 + 0.348i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4025\)    =    \(5^{2} \cdot 7 \cdot 23\)
Sign: $0.937 + 0.348i$
Analytic conductor: \(18.6920\)
Root analytic conductor: \(18.6920\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4025} (842, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4025,\ (0:\ ),\ 0.937 + 0.348i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7395371137 + 0.1331652153i\)
\(L(\frac12)\) \(\approx\) \(0.7395371137 + 0.1331652153i\)
\(L(1)\) \(\approx\) \(0.6971117199 + 0.6347203172i\)
\(L(1)\) \(\approx\) \(0.6971117199 + 0.6347203172i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.151 + 0.988i)T \)
3 \( 1 + (0.556 + 0.830i)T \)
11 \( 1 + (-0.449 + 0.893i)T \)
13 \( 1 + (-0.226 - 0.974i)T \)
17 \( 1 + (-0.976 + 0.217i)T \)
19 \( 1 + (-0.398 - 0.917i)T \)
29 \( 1 + (-0.993 + 0.113i)T \)
31 \( 1 + (-0.0665 + 0.997i)T \)
37 \( 1 + (-0.924 - 0.380i)T \)
41 \( 1 + (0.941 + 0.336i)T \)
43 \( 1 + (-0.989 - 0.142i)T \)
47 \( 1 + (-0.207 - 0.978i)T \)
53 \( 1 + (0.992 + 0.123i)T \)
59 \( 1 + (0.290 - 0.956i)T \)
61 \( 1 + (-0.272 + 0.962i)T \)
67 \( 1 + (0.353 + 0.935i)T \)
71 \( 1 + (-0.254 - 0.967i)T \)
73 \( 1 + (-0.318 + 0.948i)T \)
79 \( 1 + (0.483 - 0.875i)T \)
83 \( 1 + (0.999 - 0.0285i)T \)
89 \( 1 + (-0.532 + 0.846i)T \)
97 \( 1 + (0.999 + 0.0285i)T \)
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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

−18.581059600153015159302370482106, −18.16983793216850807174147313950, −17.225004808064567391647893341862, −16.625859613181671729973342279611, −15.506737265087039735968374903090, −14.72061839009699478128145512319, −14.10773204061440920363816012649, −13.51883968092509870522520215492, −13.02735874827092646454008731010, −12.30436956892812150034368184639, −11.55573094088794626633334377993, −11.11719940110915451530704069367, −10.17798369357882094610417751792, −9.32190750684549502725078651007, −8.82510925076643707893997983540, −8.13823110060994025447463356419, −7.380522319725193994193169392741, −6.35149082360396828486968295270, −5.79240749752888092220757249107, −4.744171599537917803967711917398, −3.87246178470428757567512990289, −3.28535887757673739492623944016, −2.251572861535006270596517280869, −1.95712290308123856585774508806, −0.86403803960823316634595723391, 0.21230559749774661358494866548, 1.948751183119644154900234949535, 2.8268450323201328454765568907, 3.65504393292558076084754545276, 4.42113765515096737046990065205, 5.06112546338891966048929829455, 5.563575901488131778053136362403, 6.719149992997989461909956067321, 7.32284244202430094460580348638, 8.06315921902174502495750332520, 8.78712433464217636421338495815, 9.251117169595623258547701694038, 10.20069111635135372752235258681, 10.5637263084501875247313341059, 11.647780496066544120574571191401, 12.82193122756548911565291089803, 13.087136363470955302141643193863, 13.89321769330024471732014154843, 14.752190682446752771263395327866, 15.183061301883099374444301156475, 15.58674201370870404576474900566, 16.29578854516030832314760716183, 17.07573933822849641184734322202, 17.72925228504336945121109955556, 18.17100329574430883623502895339

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