Properties

Label 1-15e2-225.11-r1-0-0
Degree $1$
Conductor $225$
Sign $-0.152 - 0.988i$
Analytic cond. $24.1796$
Root an. cond. $24.1796$
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  + (0.978 − 0.207i)2-s + (0.913 − 0.406i)4-s + (−0.5 − 0.866i)7-s + (0.809 − 0.587i)8-s + (0.978 − 0.207i)11-s + (−0.978 − 0.207i)13-s + (−0.669 − 0.743i)14-s + (0.669 − 0.743i)16-s + (0.809 − 0.587i)17-s + (−0.809 + 0.587i)19-s + (0.913 − 0.406i)22-s + (−0.669 − 0.743i)23-s − 26-s + (−0.809 − 0.587i)28-s + (0.104 − 0.994i)29-s + ⋯
L(s)  = 1  + (0.978 − 0.207i)2-s + (0.913 − 0.406i)4-s + (−0.5 − 0.866i)7-s + (0.809 − 0.587i)8-s + (0.978 − 0.207i)11-s + (−0.978 − 0.207i)13-s + (−0.669 − 0.743i)14-s + (0.669 − 0.743i)16-s + (0.809 − 0.587i)17-s + (−0.809 + 0.587i)19-s + (0.913 − 0.406i)22-s + (−0.669 − 0.743i)23-s − 26-s + (−0.809 − 0.587i)28-s + (0.104 − 0.994i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.152 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.152 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.152 - 0.988i$
Analytic conductor: \(24.1796\)
Root analytic conductor: \(24.1796\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 225,\ (1:\ ),\ -0.152 - 0.988i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.079867375 - 2.426622904i\)
\(L(\frac12)\) \(\approx\) \(2.079867375 - 2.426622904i\)
\(L(1)\) \(\approx\) \(1.725450926 - 0.7410134628i\)
\(L(1)\) \(\approx\) \(1.725450926 - 0.7410134628i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + (0.978 - 0.207i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (0.978 - 0.207i)T \)
13 \( 1 + (-0.978 - 0.207i)T \)
17 \( 1 + (0.809 - 0.587i)T \)
19 \( 1 + (-0.809 + 0.587i)T \)
23 \( 1 + (-0.669 - 0.743i)T \)
29 \( 1 + (0.104 - 0.994i)T \)
31 \( 1 + (-0.104 - 0.994i)T \)
37 \( 1 + (0.309 + 0.951i)T \)
41 \( 1 + (0.978 + 0.207i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (0.104 - 0.994i)T \)
53 \( 1 + (0.809 + 0.587i)T \)
59 \( 1 + (0.978 + 0.207i)T \)
61 \( 1 + (-0.978 + 0.207i)T \)
67 \( 1 + (-0.104 - 0.994i)T \)
71 \( 1 + (0.809 + 0.587i)T \)
73 \( 1 + (0.309 - 0.951i)T \)
79 \( 1 + (-0.104 + 0.994i)T \)
83 \( 1 + (-0.913 - 0.406i)T \)
89 \( 1 + (-0.309 + 0.951i)T \)
97 \( 1 + (-0.104 + 0.994i)T \)
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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

−26.04558814317968229963098419766, −25.38113262179498048182754168017, −24.57074507645615423928538555923, −23.6518613676240676564403585209, −22.66151709552166913417071181509, −21.78688696701364672747327844619, −21.352943782307830022007596311607, −19.72938892055760037021371478557, −19.42692292624579681844898971047, −17.76960603621301217128291119857, −16.764128482190616341993456661669, −15.86925119922772292306440568253, −14.80119969219575924889865974700, −14.27050450105511686564725836219, −12.77499981511755380823603530097, −12.29642480754937283439583437698, −11.28740510752199236707864534814, −9.89799425143693678310227057119, −8.72230925970402342028652289568, −7.34208514264394728946101569068, −6.366808852723630023079564976128, −5.4056190184616111020238428652, −4.196622771664384180050709237425, −3.025940330769102639761727436981, −1.81548629488483423020222120768, 0.73827793279073828582716311808, 2.340836141937146484909475866180, 3.66002168747465126274895450115, 4.47512914213940077601999101115, 5.88745980702537825719024087012, 6.80834339283774440161164570661, 7.85362926034550279942184334720, 9.68136427691814746000415410804, 10.39628673341038451564216181240, 11.68931692207078741648840103796, 12.43568064490080251865419039898, 13.54698133311428405204620144991, 14.33197071822649185388595909648, 15.1834653473888984058216794268, 16.59868375025542722066661789210, 16.937037972581675772243540672254, 18.75965052925399708517053299604, 19.66391734969329248312074977701, 20.35053340857832839566589180061, 21.37788089749229939269594176337, 22.44541066317202649861836700488, 22.92201830700765312863084136144, 24.01180264870037040563242406240, 24.83246432623591604094569956737, 25.718054738782753382946523120974

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