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Properties

Label 1-6025-6025.164-r0-0-0
Degree $1$
Conductor $6025$
Sign $0.286 - 0.958i$
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  + (0.743 − 0.669i)2-s + (0.406 − 0.913i)3-s + (0.104 − 0.994i)4-s + (−0.309 − 0.951i)6-s + (0.777 + 0.629i)7-s + (−0.587 − 0.809i)8-s + (−0.669 − 0.743i)9-s + (0.998 + 0.0523i)11-s + (−0.866 − 0.5i)12-s + (−0.258 + 0.965i)13-s + (0.998 − 0.0523i)14-s + (−0.978 − 0.207i)16-s + (−0.891 + 0.453i)17-s + (−0.994 − 0.104i)18-s + (0.777 + 0.629i)19-s + ⋯
L(s)  = 1  + (0.743 − 0.669i)2-s + (0.406 − 0.913i)3-s + (0.104 − 0.994i)4-s + (−0.309 − 0.951i)6-s + (0.777 + 0.629i)7-s + (−0.587 − 0.809i)8-s + (−0.669 − 0.743i)9-s + (0.998 + 0.0523i)11-s + (−0.866 − 0.5i)12-s + (−0.258 + 0.965i)13-s + (0.998 − 0.0523i)14-s + (−0.978 − 0.207i)16-s + (−0.891 + 0.453i)17-s + (−0.994 − 0.104i)18-s + (0.777 + 0.629i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.286 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.286 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $0.286 - 0.958i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (164, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ 0.286 - 0.958i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.982343188 - 2.221369310i\)
\(L(\frac12)\) \(\approx\) \(2.982343188 - 2.221369310i\)
\(L(1)\) \(\approx\) \(1.658724517 - 1.064940810i\)
\(L(1)\) \(\approx\) \(1.658724517 - 1.064940810i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (0.743 - 0.669i)T \)
3 \( 1 + (0.406 - 0.913i)T \)
7 \( 1 + (0.777 + 0.629i)T \)
11 \( 1 + (0.998 + 0.0523i)T \)
13 \( 1 + (-0.258 + 0.965i)T \)
17 \( 1 + (-0.891 + 0.453i)T \)
19 \( 1 + (0.777 + 0.629i)T \)
23 \( 1 + (0.987 - 0.156i)T \)
29 \( 1 + (-0.207 + 0.978i)T \)
31 \( 1 + (-0.0523 + 0.998i)T \)
37 \( 1 + (0.777 + 0.629i)T \)
41 \( 1 + (0.587 - 0.809i)T \)
43 \( 1 + (-0.156 - 0.987i)T \)
47 \( 1 + (-0.951 + 0.309i)T \)
53 \( 1 + (-0.743 - 0.669i)T \)
59 \( 1 + (0.994 - 0.104i)T \)
61 \( 1 + (-0.587 + 0.809i)T \)
67 \( 1 + (0.743 - 0.669i)T \)
71 \( 1 + (-0.544 - 0.838i)T \)
73 \( 1 + (-0.707 + 0.707i)T \)
79 \( 1 + (0.951 + 0.309i)T \)
83 \( 1 + (0.913 - 0.406i)T \)
89 \( 1 + (-0.0523 + 0.998i)T \)
97 \( 1 + (0.669 - 0.743i)T \)
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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

−17.47838987689092996055205937360, −17.13949072094634831996132134447, −16.392152508255277166457559979175, −15.78272115461988009781745140791, −15.021295019578669621889959816575, −14.742600052042602538583880279892, −14.090739613334400182915941109014, −13.340046875038622126501426736059, −13.01182119239867280014951032906, −11.605210697792785870085913277431, −11.434247249448589467608748108976, −10.75604493370349741185128482171, −9.59859345827828813578717578195, −9.24839164293334963103570646163, −8.3125668004178101801466908623, −7.76676240095474687632623083285, −7.16675061671473506600093381379, −6.244348666210755132814245828215, −5.48031574774628354713743236951, −4.65431529393300018024323359117, −4.4472571639393553019987768941, −3.54888862487472749146100027061, −2.90856433426303077488862174770, −2.10940358972090978505604854953, −0.73180940696962603012197434434, 1.00069271630418883596249494249, 1.691597020409602417834802507838, 2.075042795594091709338114993750, 3.03780303803494511345643001066, 3.698360599092027483925238529761, 4.583230517297080407569594695156, 5.23473729616360608652459838171, 6.106799677176465339531691349177, 6.715369137947793657358790762980, 7.266525073782095742768841211338, 8.35985054561136483007839676594, 9.04960323551335817015190608051, 9.35059055258334482762222547423, 10.51791980689285605980852200910, 11.40466057932116881626547314606, 11.63657986523401610117664357610, 12.38426489509954492962177536366, 12.81880888823771896365803085003, 13.7394800517738465387683821282, 14.225876232116791073901704352981, 14.69139811565453616339330948989, 15.19424345272806358421229372848, 16.18515581442227846396858301784, 17.0573809826223085647321765131, 17.88007858749827551622475388841

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