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Properties

Label 1-15e2-225.113-r0-0-0
Degree $1$
Conductor $225$
Sign $0.152 - 0.988i$
Analytic cond. $1.04489$
Root an. cond. $1.04489$
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  + (0.207 − 0.978i)2-s + (−0.913 − 0.406i)4-s + (0.866 + 0.5i)7-s + (−0.587 + 0.809i)8-s + (0.978 + 0.207i)11-s + (−0.207 − 0.978i)13-s + (0.669 − 0.743i)14-s + (0.669 + 0.743i)16-s + (0.587 − 0.809i)17-s + (0.809 + 0.587i)19-s + (0.406 − 0.913i)22-s + (−0.743 − 0.669i)23-s − 26-s + (−0.587 − 0.809i)28-s + (−0.104 − 0.994i)29-s + ⋯
L(s)  = 1  + (0.207 − 0.978i)2-s + (−0.913 − 0.406i)4-s + (0.866 + 0.5i)7-s + (−0.587 + 0.809i)8-s + (0.978 + 0.207i)11-s + (−0.207 − 0.978i)13-s + (0.669 − 0.743i)14-s + (0.669 + 0.743i)16-s + (0.587 − 0.809i)17-s + (0.809 + 0.587i)19-s + (0.406 − 0.913i)22-s + (−0.743 − 0.669i)23-s − 26-s + (−0.587 − 0.809i)28-s + (−0.104 − 0.994i)29-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.152 - 0.988i$
Analytic conductor: \(1.04489\)
Root analytic conductor: \(1.04489\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 225,\ (0:\ ),\ 0.152 - 0.988i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.020814212 - 0.8749435984i\)
\(L(\frac12)\) \(\approx\) \(1.020814212 - 0.8749435984i\)
\(L(1)\) \(\approx\) \(1.044992653 - 0.5801281925i\)
\(L(1)\) \(\approx\) \(1.044992653 - 0.5801281925i\)

Euler product

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

−26.43998096117803152645142481717, −25.80347717358667468560471410751, −24.473192394092142920412155959177, −24.11312387594570909512086237235, −23.17688559776730274333345697267, −22.00536790777919924245810423788, −21.414226622849294929713892256, −20.07897753057190584050124708682, −18.96267238404807584720775411961, −17.86221544761902358072907766457, −17.0574016280357864731968746974, −16.35571923815805062978654366680, −15.08939332015846385932948003209, −14.25248506364127536999663840366, −13.66762310218733082725201707018, −12.2485514532842406401371379660, −11.305268749305632669632307993821, −9.78090091366329224280418636873, −8.80293135158863043374220661487, −7.725529868160648689663140682583, −6.84604865151820489888758671134, −5.67733642634552028069091519135, −4.50979087627368497826729109475, −3.610564218249674115535943894159, −1.44528740838425776927695377016, 1.18987867163608393605377179324, 2.456541181201413050470962211921, 3.71467279433044519058706115147, 4.93764059498245671541320292065, 5.83974120990686123079224642228, 7.64267032486695626493725021514, 8.73606645027034079224481123563, 9.759540874409519696521176017785, 10.75457810186251362173681839262, 11.97772843964164242694910989685, 12.268032046419142785999730788092, 13.86692191683192928590578673903, 14.437112356547198254060895914903, 15.52923265265433567621045699342, 17.07815987583646443127895044632, 17.99564706681398040283720428154, 18.696632594326255089563725052715, 19.92436148586038045667097615091, 20.55436320895929771423467303257, 21.501016676991839742036108594045, 22.44592598819348369435319638724, 23.07340897641240943757016887910, 24.46376240567036594366193008641, 25.02177150392063917518123931353, 26.6019719495748285909154705537

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