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Properties

Label 1-6025-6025.2822-r0-0-0
Degree $1$
Conductor $6025$
Sign $0.533 - 0.845i$
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  + (−0.358 + 0.933i)2-s + (−0.933 + 0.358i)3-s + (−0.743 − 0.669i)4-s − i·6-s + (0.725 + 0.688i)7-s + (0.891 − 0.453i)8-s + (0.743 − 0.669i)9-s + (−0.688 + 0.725i)11-s + (0.933 + 0.358i)12-s + (0.878 + 0.477i)13-s + (−0.902 + 0.430i)14-s + (0.104 + 0.994i)16-s + (−0.233 − 0.972i)17-s + (0.358 + 0.933i)18-s + (−0.430 + 0.902i)19-s + ⋯
L(s)  = 1  + (−0.358 + 0.933i)2-s + (−0.933 + 0.358i)3-s + (−0.743 − 0.669i)4-s − i·6-s + (0.725 + 0.688i)7-s + (0.891 − 0.453i)8-s + (0.743 − 0.669i)9-s + (−0.688 + 0.725i)11-s + (0.933 + 0.358i)12-s + (0.878 + 0.477i)13-s + (−0.902 + 0.430i)14-s + (0.104 + 0.994i)16-s + (−0.233 − 0.972i)17-s + (0.358 + 0.933i)18-s + (−0.430 + 0.902i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.533 - 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.533 - 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $0.533 - 0.845i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (2822, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ 0.533 - 0.845i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1140265545 - 0.06286791707i\)
\(L(\frac12)\) \(\approx\) \(0.1140265545 - 0.06286791707i\)
\(L(1)\) \(\approx\) \(0.4936671236 + 0.3360186835i\)
\(L(1)\) \(\approx\) \(0.4936671236 + 0.3360186835i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (-0.358 + 0.933i)T \)
3 \( 1 + (-0.933 + 0.358i)T \)
7 \( 1 + (0.725 + 0.688i)T \)
11 \( 1 + (-0.688 + 0.725i)T \)
13 \( 1 + (0.878 + 0.477i)T \)
17 \( 1 + (-0.233 - 0.972i)T \)
19 \( 1 + (-0.430 + 0.902i)T \)
23 \( 1 + (0.996 - 0.0784i)T \)
29 \( 1 + (0.358 - 0.933i)T \)
31 \( 1 + (-0.284 + 0.958i)T \)
37 \( 1 + (0.182 + 0.983i)T \)
41 \( 1 + (-0.707 - 0.707i)T \)
43 \( 1 + (-0.972 + 0.233i)T \)
47 \( 1 + (-0.156 - 0.987i)T \)
53 \( 1 + (-0.629 - 0.777i)T \)
59 \( 1 + (-0.544 - 0.838i)T \)
61 \( 1 + (0.453 + 0.891i)T \)
67 \( 1 + (-0.777 - 0.629i)T \)
71 \( 1 + (0.999 - 0.0261i)T \)
73 \( 1 + (-0.852 - 0.522i)T \)
79 \( 1 + (-0.707 - 0.707i)T \)
83 \( 1 + (-0.913 + 0.406i)T \)
89 \( 1 + (0.999 - 0.0261i)T \)
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

−17.875077485546057521834399087017, −17.26874626309261557432534711465, −16.85300719122117131187346083625, −16.09464081559776104095402395121, −15.30178873308008655935792369276, −14.32936108126929633142738379495, −13.455845861623544035590542040134, −13.07599150998806579766847592115, −12.67219859426680097071203898081, −11.49822099600222149815032108165, −11.22044554234011064733020412719, −10.607631262673979060061664711286, −10.370064198649444519594478877369, −9.16747946171905029866542584201, −8.42550209175787838278814079740, −7.90023782942832832866139896182, −7.18496389985652032818053201138, −6.31441988689663605968608255496, −5.44723968496608822969310321366, −4.81455829741410304757018661271, −4.12908970979091790502500506328, −3.30641827355348763615626902910, −2.404989724684966738416890406064, −1.4187713241253083832112637087, −0.98567148559896936416149411342, 0.05135522012358153032530142143, 1.30452366701407155534275309862, 1.93141533901494237584165292958, 3.29147673611943081378545558254, 4.35701077529500612105166860085, 4.89781933567432978327975002783, 5.326327312452547894378730627796, 6.145445820553308861921192906699, 6.74145721514384973970964521137, 7.4047081426144385142398973285, 8.31348027083966989886373030004, 8.79472726086093559194522356677, 9.65289256304202050974448357438, 10.204841866498085423747752558678, 10.91941024845877353759256427308, 11.58495597137216652775584485943, 12.259262875426082908009523311401, 13.09121560961062304310640802832, 13.749384714781914978372771053853, 14.66010555556133881943109402672, 15.19732547362548990306626562992, 15.6903325356992954416256142235, 16.2886917381894587492052384863, 16.92644291676823999998543174152, 17.58035654251288047149357206083

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