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Properties

Label 1-6025-6025.762-r0-0-0
Degree $1$
Conductor $6025$
Sign $-0.257 - 0.966i$
Analytic cond. $27.9799$
Root an. cond. $27.9799$
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  + (−0.0523 − 0.998i)2-s + (0.358 − 0.933i)3-s + (−0.994 + 0.104i)4-s + (−0.951 − 0.309i)6-s + (0.284 + 0.958i)7-s + (0.156 + 0.987i)8-s + (−0.743 − 0.669i)9-s + (0.942 + 0.333i)11-s + (−0.258 + 0.965i)12-s + (−0.793 + 0.608i)13-s + (0.942 − 0.333i)14-s + (0.978 − 0.207i)16-s + (0.0784 − 0.996i)17-s + (−0.629 + 0.777i)18-s + (0.284 + 0.958i)19-s + ⋯
L(s)  = 1  + (−0.0523 − 0.998i)2-s + (0.358 − 0.933i)3-s + (−0.994 + 0.104i)4-s + (−0.951 − 0.309i)6-s + (0.284 + 0.958i)7-s + (0.156 + 0.987i)8-s + (−0.743 − 0.669i)9-s + (0.942 + 0.333i)11-s + (−0.258 + 0.965i)12-s + (−0.793 + 0.608i)13-s + (0.942 − 0.333i)14-s + (0.978 − 0.207i)16-s + (0.0784 − 0.996i)17-s + (−0.629 + 0.777i)18-s + (0.284 + 0.958i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.257 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.257 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $-0.257 - 0.966i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (762, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ -0.257 - 0.966i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9752936996 - 1.269778685i\)
\(L(\frac12)\) \(\approx\) \(0.9752936996 - 1.269778685i\)
\(L(1)\) \(\approx\) \(0.8371296142 - 0.6293467513i\)
\(L(1)\) \(\approx\) \(0.8371296142 - 0.6293467513i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (-0.0523 - 0.998i)T \)
3 \( 1 + (0.358 - 0.933i)T \)
7 \( 1 + (0.284 + 0.958i)T \)
11 \( 1 + (0.942 + 0.333i)T \)
13 \( 1 + (-0.793 + 0.608i)T \)
17 \( 1 + (0.0784 - 0.996i)T \)
19 \( 1 + (0.284 + 0.958i)T \)
23 \( 1 + (-0.522 + 0.852i)T \)
29 \( 1 + (0.544 - 0.838i)T \)
31 \( 1 + (-0.430 - 0.902i)T \)
37 \( 1 + (-0.284 - 0.958i)T \)
41 \( 1 + (-0.987 - 0.156i)T \)
43 \( 1 + (-0.233 - 0.972i)T \)
47 \( 1 + (-0.891 + 0.453i)T \)
53 \( 1 + (0.998 + 0.0523i)T \)
59 \( 1 + (0.777 - 0.629i)T \)
61 \( 1 + (-0.156 + 0.987i)T \)
67 \( 1 + (0.0523 + 0.998i)T \)
71 \( 1 + (0.182 - 0.983i)T \)
73 \( 1 + (0.923 - 0.382i)T \)
79 \( 1 + (-0.453 + 0.891i)T \)
83 \( 1 + (-0.913 - 0.406i)T \)
89 \( 1 + (0.430 + 0.902i)T \)
97 \( 1 + (-0.669 - 0.743i)T \)
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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

−17.56112341651371558547438950955, −17.06513777298222430906527248737, −16.60324907942539525870309444833, −16.00895233235030768209736553224, −15.1974871528793361432959466551, −14.646071210609205570639002095973, −14.287423509138966151249882513946, −13.59621558145461555324414319321, −12.91822317379953530541656820680, −11.99494243642660769358611448875, −11.05847077644598533163171492113, −10.30524989044786011828160342420, −9.975223514031788470487469993584, −9.13065993067667826033995378627, −8.37000923171602773060447479792, −8.110581856328388591042431462409, −7.00315087906825282187162619797, −6.62843158550022317769253419652, −5.6022345723401223125372687061, −4.84607751684684223826728324994, −4.46712157078482658512099385256, −3.58699047696638944405646612715, −3.11429967645774191236246143354, −1.701739223034676091942997525, −0.66117533338139497358971741339, 0.58814637713833197982723589814, 1.75997956400041643961284989913, 1.957206962362259337749703992644, 2.768987485578454169605406432524, 3.596107297047659602642357638910, 4.31944118625429798329905227234, 5.337751866826607098609310588508, 5.83516861973635094012622980911, 6.87967223857890538915304697180, 7.56038124018431155851199106644, 8.2870397814293434848641029018, 8.95171526034339618171716279945, 9.56010890591631401883117954856, 9.97281338586948180424677394879, 11.38992540896504889025025968100, 11.69329089243001132078284541891, 12.14223088023919029035166820907, 12.65021513944919254192852187512, 13.64306175235359125732940608904, 14.04423337120243564587598374787, 14.6513484458925627118389590322, 15.251660449941548422147336864776, 16.4344235522494391003254180403, 17.16288989844931338403259069609, 17.79221474104510912051477741823

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