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Properties

Label 1-4025-4025.1133-r0-0-0
Degree $1$
Conductor $4025$
Sign $0.279 + 0.960i$
Analytic cond. $18.6920$
Root an. cond. $18.6920$
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  + (0.226 − 0.974i)2-s + (−0.633 − 0.774i)3-s + (−0.897 − 0.441i)4-s + (−0.897 + 0.441i)6-s + (−0.633 + 0.774i)8-s + (−0.198 + 0.980i)9-s + (0.0855 − 0.996i)11-s + (0.226 + 0.974i)12-s + (0.336 − 0.941i)13-s + (0.610 + 0.791i)16-s + (0.441 + 0.897i)17-s + (0.909 + 0.415i)18-s + (−0.985 + 0.170i)19-s + (−0.951 − 0.309i)22-s + 24-s + ⋯
L(s)  = 1  + (0.226 − 0.974i)2-s + (−0.633 − 0.774i)3-s + (−0.897 − 0.441i)4-s + (−0.897 + 0.441i)6-s + (−0.633 + 0.774i)8-s + (−0.198 + 0.980i)9-s + (0.0855 − 0.996i)11-s + (0.226 + 0.974i)12-s + (0.336 − 0.941i)13-s + (0.610 + 0.791i)16-s + (0.441 + 0.897i)17-s + (0.909 + 0.415i)18-s + (−0.985 + 0.170i)19-s + (−0.951 − 0.309i)22-s + 24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.279 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.279 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4025\)    =    \(5^{2} \cdot 7 \cdot 23\)
Sign: $0.279 + 0.960i$
Analytic conductor: \(18.6920\)
Root analytic conductor: \(18.6920\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4025} (1133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4025,\ (0:\ ),\ 0.279 + 0.960i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2829762658 - 0.2123220158i\)
\(L(\frac12)\) \(\approx\) \(-0.2829762658 - 0.2123220158i\)
\(L(1)\) \(\approx\) \(0.4861918486 - 0.5694877705i\)
\(L(1)\) \(\approx\) \(0.4861918486 - 0.5694877705i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.226 - 0.974i)T \)
3 \( 1 + (-0.633 - 0.774i)T \)
11 \( 1 + (0.0855 - 0.996i)T \)
13 \( 1 + (0.336 - 0.941i)T \)
17 \( 1 + (0.441 + 0.897i)T \)
19 \( 1 + (-0.985 + 0.170i)T \)
29 \( 1 + (0.985 + 0.170i)T \)
31 \( 1 + (-0.774 - 0.633i)T \)
37 \( 1 + (-0.980 - 0.198i)T \)
41 \( 1 + (0.870 - 0.491i)T \)
43 \( 1 + (-0.540 + 0.841i)T \)
47 \( 1 + (0.951 + 0.309i)T \)
53 \( 1 + (-0.825 - 0.564i)T \)
59 \( 1 + (0.941 + 0.336i)T \)
61 \( 1 + (0.362 - 0.931i)T \)
67 \( 1 + (-0.856 - 0.516i)T \)
71 \( 1 + (-0.921 + 0.389i)T \)
73 \( 1 + (-0.884 + 0.466i)T \)
79 \( 1 + (0.0285 - 0.999i)T \)
83 \( 1 + (0.676 + 0.736i)T \)
89 \( 1 + (-0.998 + 0.0570i)T \)
97 \( 1 + (0.676 - 0.736i)T \)
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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

−18.72681734179460924145379342379, −18.067838607577160673966420247539, −17.40682935171372829274914486675, −16.95482372443680207319528133999, −16.07831662125472443476760731150, −15.87723737203628294234475474608, −14.94628239777043190757959747670, −14.50403996821738075106556425511, −13.77790747088455405371896280592, −12.87653226660881472563536605351, −12.10130882255350823669719846174, −11.69676623704470448414556041690, −10.53428512670525396614123341404, −10.02628074302085412272195452758, −9.03242063849730209982067055098, −8.894196848333539431165648121796, −7.65761505420940920338692430286, −6.87288895794051438933249437292, −6.45666807644206163622854139564, −5.52733895090319092670140849238, −4.868108818358449316927413932514, −4.29889510458575704044819985764, −3.66768116762645774175794749635, −2.59315462574580939286944583363, −1.23438186510632498401442262366, 0.119743894253682090897507219328, 1.03561028291665965759578839638, 1.75713975778543452751601334143, 2.67725032822659308697058586424, 3.44760675366946233282043872546, 4.27162989047586826069438549399, 5.25077809792912811475857314065, 5.88863335619180304656105367863, 6.312811031005705527058681227686, 7.528548020460077641423938228, 8.36920749949782393029198989721, 8.701224500862948771201177179114, 9.96014947972992965927820023762, 10.64286375911133114460848839399, 11.00602845626806732997650138111, 11.7555628790120405228702724261, 12.654127994705878991684589274517, 12.793571088385237860616819216180, 13.63675372851303722422851621647, 14.262138386886342584039785321739, 15.00765362895571587643696917028, 16.00891832973467205129879056579, 16.79679732219754288447161413975, 17.53192800976788963200463168148, 17.937191566984592980386039572278

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