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Properties

Label 1-75e2-5625.661-r0-0-0
Degree $1$
Conductor $5625$
Sign $-0.276 + 0.961i$
Analytic cond. $26.1223$
Root an. cond. $26.1223$
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  + (0.0961 − 0.995i)2-s + (−0.981 − 0.191i)4-s + (0.604 − 0.796i)7-s + (−0.285 + 0.958i)8-s + (0.976 − 0.216i)11-s + (−0.220 − 0.975i)13-s + (−0.734 − 0.678i)14-s + (0.926 + 0.375i)16-s + (−0.0878 − 0.996i)17-s + (−0.888 − 0.459i)19-s + (−0.121 − 0.992i)22-s + (0.630 − 0.775i)23-s + (−0.992 + 0.125i)26-s + (−0.745 + 0.666i)28-s + (−0.410 + 0.911i)29-s + ⋯
L(s)  = 1  + (0.0961 − 0.995i)2-s + (−0.981 − 0.191i)4-s + (0.604 − 0.796i)7-s + (−0.285 + 0.958i)8-s + (0.976 − 0.216i)11-s + (−0.220 − 0.975i)13-s + (−0.734 − 0.678i)14-s + (0.926 + 0.375i)16-s + (−0.0878 − 0.996i)17-s + (−0.888 − 0.459i)19-s + (−0.121 − 0.992i)22-s + (0.630 − 0.775i)23-s + (−0.992 + 0.125i)26-s + (−0.745 + 0.666i)28-s + (−0.410 + 0.911i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.276 + 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.276 + 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5625\)    =    \(3^{2} \cdot 5^{4}\)
Sign: $-0.276 + 0.961i$
Analytic conductor: \(26.1223\)
Root analytic conductor: \(26.1223\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5625} (661, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5625,\ (0:\ ),\ -0.276 + 0.961i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.4518849839 - 0.6001945439i\)
\(L(\frac12)\) \(\approx\) \(-0.4518849839 - 0.6001945439i\)
\(L(1)\) \(\approx\) \(0.6702963080 - 0.6571602030i\)
\(L(1)\) \(\approx\) \(0.6702963080 - 0.6571602030i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + (0.0961 - 0.995i)T \)
7 \( 1 + (0.604 - 0.796i)T \)
11 \( 1 + (0.976 - 0.216i)T \)
13 \( 1 + (-0.220 - 0.975i)T \)
17 \( 1 + (-0.0878 - 0.996i)T \)
19 \( 1 + (-0.888 - 0.459i)T \)
23 \( 1 + (0.630 - 0.775i)T \)
29 \( 1 + (-0.410 + 0.911i)T \)
31 \( 1 + (-0.818 + 0.574i)T \)
37 \( 1 + (0.693 + 0.720i)T \)
41 \( 1 + (-0.455 - 0.890i)T \)
43 \( 1 + (-0.895 - 0.444i)T \)
47 \( 1 + (0.0795 + 0.996i)T \)
53 \( 1 + (-0.962 - 0.272i)T \)
59 \( 1 + (-0.941 - 0.336i)T \)
61 \( 1 + (-0.455 + 0.890i)T \)
67 \( 1 + (-0.203 + 0.979i)T \)
71 \( 1 + (-0.999 - 0.0251i)T \)
73 \( 1 + (-0.236 - 0.971i)T \)
79 \( 1 + (-0.935 - 0.352i)T \)
83 \( 1 + (-0.364 + 0.931i)T \)
89 \( 1 + (0.850 - 0.525i)T \)
97 \( 1 + (0.868 - 0.496i)T \)
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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

−18.23706888088541197332011688380, −17.34941006334714305410512821777, −17.00305519574626824737932562559, −16.47521175366271709861610280839, −15.480787581860438146846183353090, −14.858278265787355045927267753468, −14.70864240283324286754463321521, −13.88167900363447972251791396618, −13.06411984946784559273254311627, −12.48699293051786470414457245041, −11.701049462042894121915223718898, −11.167261548742841837834843219789, −10.03988780872352843943370380864, −9.26807424719479982856142935902, −8.93637832117043434206364756593, −8.11016575389116284944780974392, −7.55376246784747225220142115315, −6.61065328165714539477225746937, −6.14989132805851545617301730107, −5.48513003511847146910776529202, −4.52656358438676144619546534177, −4.150721094896024703115498916922, −3.26792770712409984610375651062, −1.96547729001396027934193735400, −1.47674439158739709718663793411, 0.19220829603861900978340531546, 1.10384315812094008248192872367, 1.74370557794763570927843920468, 2.817751366561128530184879227744, 3.350568469359075294276501452613, 4.26833800020360834538226333748, 4.783600605269364106362452519436, 5.45583888448063542315447415629, 6.51289962258368088757451152211, 7.25329927454832974400766139261, 8.06644459088001692129417578014, 8.87201336498252416302513601398, 9.271224701227823533712107244744, 10.37300568156124643204421158449, 10.632218581037522325849824122215, 11.375226301681412337528953704687, 11.93239685371124098541153135237, 12.79467105647167604917828962546, 13.239450621470895451290560387170, 14.05066678160104700538801128668, 14.58044558054691291914560135418, 15.05859810362049140282431206532, 16.20525506876770621538019257561, 17.03105670734076883119931588203, 17.363028883110600471095115801106

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