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Properties

Label 1-6025-6025.3038-r0-0-0
Degree $1$
Conductor $6025$
Sign $0.783 + 0.620i$
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  + (−0.998 + 0.0523i)2-s + (0.0523 − 0.998i)3-s + (0.994 − 0.104i)4-s + i·6-s + (0.430 + 0.902i)7-s + (−0.987 + 0.156i)8-s + (−0.994 − 0.104i)9-s + (0.902 − 0.430i)11-s + (−0.0523 − 0.998i)12-s + (0.182 + 0.983i)13-s + (−0.477 − 0.878i)14-s + (0.978 − 0.207i)16-s + (0.996 − 0.0784i)17-s + (0.998 + 0.0523i)18-s + (−0.878 − 0.477i)19-s + ⋯
L(s)  = 1  + (−0.998 + 0.0523i)2-s + (0.0523 − 0.998i)3-s + (0.994 − 0.104i)4-s + i·6-s + (0.430 + 0.902i)7-s + (−0.987 + 0.156i)8-s + (−0.994 − 0.104i)9-s + (0.902 − 0.430i)11-s + (−0.0523 − 0.998i)12-s + (0.182 + 0.983i)13-s + (−0.477 − 0.878i)14-s + (0.978 − 0.207i)16-s + (0.996 − 0.0784i)17-s + (0.998 + 0.0523i)18-s + (−0.878 − 0.477i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.783 + 0.620i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.783 + 0.620i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $0.783 + 0.620i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (3038, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ 0.783 + 0.620i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.132140460 + 0.3939707584i\)
\(L(\frac12)\) \(\approx\) \(1.132140460 + 0.3939707584i\)
\(L(1)\) \(\approx\) \(0.7992794569 - 0.06267255627i\)
\(L(1)\) \(\approx\) \(0.7992794569 - 0.06267255627i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (-0.998 + 0.0523i)T \)
3 \( 1 + (0.0523 - 0.998i)T \)
7 \( 1 + (0.430 + 0.902i)T \)
11 \( 1 + (0.902 - 0.430i)T \)
13 \( 1 + (0.182 + 0.983i)T \)
17 \( 1 + (0.996 - 0.0784i)T \)
19 \( 1 + (-0.878 - 0.477i)T \)
23 \( 1 + (0.852 - 0.522i)T \)
29 \( 1 + (0.998 - 0.0523i)T \)
31 \( 1 + (-0.824 + 0.566i)T \)
37 \( 1 + (0.725 - 0.688i)T \)
41 \( 1 + (-0.707 + 0.707i)T \)
43 \( 1 + (0.0784 + 0.996i)T \)
47 \( 1 + (0.891 + 0.453i)T \)
53 \( 1 + (0.544 + 0.838i)T \)
59 \( 1 + (0.933 - 0.358i)T \)
61 \( 1 + (0.156 + 0.987i)T \)
67 \( 1 + (-0.838 - 0.544i)T \)
71 \( 1 + (-0.333 + 0.942i)T \)
73 \( 1 + (0.760 + 0.649i)T \)
79 \( 1 + (-0.707 + 0.707i)T \)
83 \( 1 + (-0.669 + 0.743i)T \)
89 \( 1 + (-0.333 + 0.942i)T \)
97 \( 1 + (0.978 + 0.207i)T \)
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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

−17.40505450045946242816559694679, −16.94477337999933449237071115534, −16.66701985494114766726702502355, −15.755873132551528136999107643398, −15.015913196717184857200053504015, −14.73702750957889178219299191212, −13.93150288388283784203494173075, −12.94731987377166131022637319602, −12.06975609968230924004243599415, −11.47261847235121650117450778292, −10.78215028039240947257722311535, −10.12201844915885268237718637344, −9.98236043958875007478274242715, −8.8946173878640038943441978450, −8.502434458541605548626658942883, −7.67617029269246013710409262259, −7.10968107110881695220513379606, −6.165243826615289227599646821463, −5.483361090552513961906362251338, −4.60039081093392721407126553865, −3.67956810265472726211932759131, −3.336297714475173912886944508691, −2.22041202854504215710326060217, −1.32480988018617740614453542762, −0.47707944670029173049671377446, 1.06794750872842589166499018533, 1.36615967940262235365276440340, 2.43364526201041408257187215093, 2.78661380795847775939457833892, 3.92672111622886501485266599278, 5.1141851165932282261982619781, 5.934579811821606600549438757460, 6.4801132320778712227374914554, 7.0166247657307928226964397398, 7.82812428320522708566311224014, 8.58308350672744025291458532799, 8.88796922529490862411627196105, 9.48895597328674100399455340649, 10.60834104556897243355160525600, 11.32396372475234743230345810271, 11.71032287166657633261422319429, 12.3541899168810879521924067644, 12.94529632784316393613371172797, 14.11029518099359303719481225453, 14.5160833770947731114825576950, 15.06315399173768164073738555411, 16.08555209118543604065819925611, 16.695938338091964492882340210787, 17.1796679743720407448585005176, 17.902282883182346010147277938456

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