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Properties

Label 1-35e2-1225.37-r1-0-0
Degree $1$
Conductor $1225$
Sign $-0.475 - 0.879i$
Analytic cond. $131.644$
Root an. cond. $131.644$
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  + (−0.0598 + 0.998i)2-s + (0.850 − 0.525i)3-s + (−0.992 − 0.119i)4-s + (0.473 + 0.880i)6-s + (0.178 − 0.983i)8-s + (0.447 − 0.894i)9-s + (−0.447 − 0.894i)11-s + (−0.907 + 0.420i)12-s + (−0.351 − 0.936i)13-s + (0.971 + 0.237i)16-s + (0.800 − 0.599i)17-s + (0.866 + 0.5i)18-s + (0.104 − 0.994i)19-s + (0.919 − 0.393i)22-s + (0.817 − 0.575i)23-s + (−0.365 − 0.930i)24-s + ⋯
L(s)  = 1  + (−0.0598 + 0.998i)2-s + (0.850 − 0.525i)3-s + (−0.992 − 0.119i)4-s + (0.473 + 0.880i)6-s + (0.178 − 0.983i)8-s + (0.447 − 0.894i)9-s + (−0.447 − 0.894i)11-s + (−0.907 + 0.420i)12-s + (−0.351 − 0.936i)13-s + (0.971 + 0.237i)16-s + (0.800 − 0.599i)17-s + (0.866 + 0.5i)18-s + (0.104 − 0.994i)19-s + (0.919 − 0.393i)22-s + (0.817 − 0.575i)23-s + (−0.365 − 0.930i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.475 - 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.475 - 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1225\)    =    \(5^{2} \cdot 7^{2}\)
Sign: $-0.475 - 0.879i$
Analytic conductor: \(131.644\)
Root analytic conductor: \(131.644\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1225} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1225,\ (1:\ ),\ -0.475 - 0.879i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8513914119 - 1.427759835i\)
\(L(\frac12)\) \(\approx\) \(0.8513914119 - 1.427759835i\)
\(L(1)\) \(\approx\) \(1.149003214 - 0.05505961827i\)
\(L(1)\) \(\approx\) \(1.149003214 - 0.05505961827i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.0598 + 0.998i)T \)
3 \( 1 + (0.850 - 0.525i)T \)
11 \( 1 + (-0.447 - 0.894i)T \)
13 \( 1 + (-0.351 - 0.936i)T \)
17 \( 1 + (0.800 - 0.599i)T \)
19 \( 1 + (0.104 - 0.994i)T \)
23 \( 1 + (0.817 - 0.575i)T \)
29 \( 1 + (-0.753 - 0.657i)T \)
31 \( 1 + (0.913 - 0.406i)T \)
37 \( 1 + (-0.907 + 0.420i)T \)
41 \( 1 + (0.134 + 0.990i)T \)
43 \( 1 + (-0.433 + 0.900i)T \)
47 \( 1 + (-0.967 + 0.251i)T \)
53 \( 1 + (0.119 - 0.992i)T \)
59 \( 1 + (0.280 + 0.959i)T \)
61 \( 1 + (-0.873 - 0.486i)T \)
67 \( 1 + (0.406 + 0.913i)T \)
71 \( 1 + (-0.393 - 0.919i)T \)
73 \( 1 + (0.635 + 0.772i)T \)
79 \( 1 + (-0.913 - 0.406i)T \)
83 \( 1 + (0.266 + 0.963i)T \)
89 \( 1 + (0.163 + 0.986i)T \)
97 \( 1 + (-0.587 - 0.809i)T \)
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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

−21.171644506093213981065700512871, −20.49382262050488836703100419835, −19.76586860650037312462736882388, −18.97370875072418137509363948545, −18.59551163073077482355272936898, −17.370101165767669777134822707, −16.74150289163007136053120152298, −15.67271484095577568998988062117, −14.74868515414048281834660904735, −14.23938876529936748140710563011, −13.41844107389730130959606826356, −12.54754463901724425215853900889, −11.93672240228692960672633513898, −10.75146624952723637341225853700, −10.18687486321408543247959911419, −9.489371293364651412546438608252, −8.79256287812778601331859848894, −7.898378380833595364797280714896, −7.12801660615470112860266726702, −5.46301175486467663662920168926, −4.74722260001776256759570623052, −3.83146693305760312161524312000, −3.19239895565117207136932081082, −2.06234030721883198689397902550, −1.51244277647531226333340597802, 0.31706018660302005932184010850, 1.09318797871496907077904867166, 2.76094447015573190518568215104, 3.30512160697066065832000240440, 4.60376350318902996939691427803, 5.45314581150447607839142412144, 6.403440535973756661222502382291, 7.2079315075162824666702670047, 8.005284884720750554316421025411, 8.45373343835270575829619086792, 9.45252518348071170142270621586, 10.07603354412641481582036652536, 11.334185681163577573658627682647, 12.50060626842030112419859576088, 13.27813145415350631708399788435, 13.66149489373344656072071654819, 14.66871689157750921996245677559, 15.1598840850927378553493284142, 15.9587200931999081508830270684, 16.81079561237613604349703286140, 17.69286588424770625764082489741, 18.362560178066583187164991561568, 19.04987948675026198200155994010, 19.65125166223507246127090357809, 20.752105420792686201289369408041

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