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Properties

Label 1-6025-6025.3169-r0-0-0
Degree $1$
Conductor $6025$
Sign $0.0286 + 0.999i$
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  + (0.809 − 0.587i)2-s − 3-s + (0.309 − 0.951i)4-s + (−0.809 + 0.587i)6-s + (−0.809 − 0.587i)7-s + (−0.309 − 0.951i)8-s + 9-s + (0.809 − 0.587i)11-s + (−0.309 + 0.951i)12-s + (0.309 − 0.951i)13-s − 14-s + (−0.809 − 0.587i)16-s + (0.309 − 0.951i)17-s + (0.809 − 0.587i)18-s + (−0.309 − 0.951i)19-s + ⋯
L(s)  = 1  + (0.809 − 0.587i)2-s − 3-s + (0.309 − 0.951i)4-s + (−0.809 + 0.587i)6-s + (−0.809 − 0.587i)7-s + (−0.309 − 0.951i)8-s + 9-s + (0.809 − 0.587i)11-s + (−0.309 + 0.951i)12-s + (0.309 − 0.951i)13-s − 14-s + (−0.809 − 0.587i)16-s + (0.309 − 0.951i)17-s + (0.809 − 0.587i)18-s + (−0.309 − 0.951i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0286 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0286 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $0.0286 + 0.999i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (3169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ 0.0286 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.4649673912 - 0.4518360447i\)
\(L(\frac12)\) \(\approx\) \(-0.4649673912 - 0.4518360447i\)
\(L(1)\) \(\approx\) \(0.7365026024 - 0.6615715721i\)
\(L(1)\) \(\approx\) \(0.7365026024 - 0.6615715721i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (0.809 - 0.587i)T \)
3 \( 1 - T \)
7 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 + (0.809 - 0.587i)T \)
13 \( 1 + (0.309 - 0.951i)T \)
17 \( 1 + (0.309 - 0.951i)T \)
19 \( 1 + (-0.309 - 0.951i)T \)
23 \( 1 + (-0.809 - 0.587i)T \)
29 \( 1 + (0.309 + 0.951i)T \)
31 \( 1 + (-0.309 + 0.951i)T \)
37 \( 1 + (-0.809 + 0.587i)T \)
41 \( 1 + (0.309 + 0.951i)T \)
43 \( 1 + (-0.809 + 0.587i)T \)
47 \( 1 + (-0.309 - 0.951i)T \)
53 \( 1 + (-0.309 - 0.951i)T \)
59 \( 1 + (-0.809 + 0.587i)T \)
61 \( 1 + (-0.809 - 0.587i)T \)
67 \( 1 + (-0.309 + 0.951i)T \)
71 \( 1 + (-0.309 - 0.951i)T \)
73 \( 1 + (0.309 + 0.951i)T \)
79 \( 1 + (-0.809 - 0.587i)T \)
83 \( 1 - T \)
89 \( 1 + (0.809 - 0.587i)T \)
97 \( 1 + (-0.309 - 0.951i)T \)
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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

−17.978635956538972408366257634619, −17.24565451918189487613245965268, −16.81732075811479340243795245080, −16.28060716804366090028975943026, −15.52945206479300019563996373203, −15.16807882233105291463226559920, −14.23432542591241483097448243919, −13.65630961214126569233432254958, −12.75465257796194317642031050121, −12.29291343413963449246874546888, −11.92421602008391279743167952678, −11.21291151037736351230897307517, −10.30486464552802649155863803364, −9.56565686589898199158093275508, −8.89099012687162373882100475433, −7.91065610793505173917434499982, −7.212509699554324957121531840943, −6.42833816905460005386790840452, −6.04137533695971052612190463028, −5.62135363019482141979703502863, −4.50281951839739341858995909226, −4.00641552651075997648449393038, −3.4650030364342588552867344670, −2.096831048683361986148712461717, −1.6254801434061547023359853964, 0.155050427121417140033809411164, 0.9244609800445526718997036062, 1.63741437063971532469108918541, 3.02799537839338220648803456980, 3.31310524786653788725583170091, 4.27277348991195104088150592577, 4.90636588077965981394191933073, 5.56499185772670673059737625277, 6.47508100747417244672791289131, 6.61717941704468649889530112564, 7.4860191115155990541149940894, 8.72294019111820808715732611330, 9.51355619736599001186185428727, 10.3692109526902876776266443440, 10.48712007944454292639427360778, 11.517115614556195286875633175490, 11.77786637373246211527153248176, 12.78704206608762388597956444260, 12.98327515411111704072864830333, 13.83748690554160620294004575102, 14.377526996078586638823611877144, 15.355358261358145630489857007389, 15.970275782930452346361777302455, 16.41109068948071521363795049299, 17.09235334513211445792669251983

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