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Properties

Label 1-1275-1275.56-r0-0-0
Degree $1$
Conductor $1275$
Sign $0.914 - 0.403i$
Analytic cond. $5.92107$
Root an. cond. $5.92107$
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  + (−0.156 + 0.987i)2-s + (−0.951 − 0.309i)4-s + (−0.923 + 0.382i)7-s + (0.453 − 0.891i)8-s + (0.852 + 0.522i)11-s + (0.587 − 0.809i)13-s + (−0.233 − 0.972i)14-s + (0.809 + 0.587i)16-s + (−0.891 − 0.453i)19-s + (−0.649 + 0.760i)22-s + (−0.852 − 0.522i)23-s + (0.707 + 0.707i)26-s + (0.996 − 0.0784i)28-s + (−0.649 + 0.760i)29-s + (0.996 + 0.0784i)31-s + (−0.707 + 0.707i)32-s + ⋯
L(s)  = 1  + (−0.156 + 0.987i)2-s + (−0.951 − 0.309i)4-s + (−0.923 + 0.382i)7-s + (0.453 − 0.891i)8-s + (0.852 + 0.522i)11-s + (0.587 − 0.809i)13-s + (−0.233 − 0.972i)14-s + (0.809 + 0.587i)16-s + (−0.891 − 0.453i)19-s + (−0.649 + 0.760i)22-s + (−0.852 − 0.522i)23-s + (0.707 + 0.707i)26-s + (0.996 − 0.0784i)28-s + (−0.649 + 0.760i)29-s + (0.996 + 0.0784i)31-s + (−0.707 + 0.707i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1275 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.914 - 0.403i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1275 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.914 - 0.403i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1275\)    =    \(3 \cdot 5^{2} \cdot 17\)
Sign: $0.914 - 0.403i$
Analytic conductor: \(5.92107\)
Root analytic conductor: \(5.92107\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1275} (56, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1275,\ (0:\ ),\ 0.914 - 0.403i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7242338796 - 0.1528004856i\)
\(L(\frac12)\) \(\approx\) \(0.7242338796 - 0.1528004856i\)
\(L(1)\) \(\approx\) \(0.7214116493 + 0.2519070505i\)
\(L(1)\) \(\approx\) \(0.7214116493 + 0.2519070505i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
17 \( 1 \)
good2 \( 1 + (-0.156 + 0.987i)T \)
7 \( 1 + (-0.923 + 0.382i)T \)
11 \( 1 + (0.852 + 0.522i)T \)
13 \( 1 + (0.587 - 0.809i)T \)
19 \( 1 + (-0.891 - 0.453i)T \)
23 \( 1 + (-0.852 - 0.522i)T \)
29 \( 1 + (-0.649 + 0.760i)T \)
31 \( 1 + (0.996 + 0.0784i)T \)
37 \( 1 + (0.852 - 0.522i)T \)
41 \( 1 + (-0.972 - 0.233i)T \)
43 \( 1 + (-0.707 - 0.707i)T \)
47 \( 1 + (-0.951 - 0.309i)T \)
53 \( 1 + (-0.453 - 0.891i)T \)
59 \( 1 + (0.156 + 0.987i)T \)
61 \( 1 + (-0.522 + 0.852i)T \)
67 \( 1 + (-0.309 - 0.951i)T \)
71 \( 1 + (-0.760 - 0.649i)T \)
73 \( 1 + (0.972 - 0.233i)T \)
79 \( 1 + (0.996 - 0.0784i)T \)
83 \( 1 + (-0.891 - 0.453i)T \)
89 \( 1 + (-0.587 - 0.809i)T \)
97 \( 1 + (0.649 - 0.760i)T \)
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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

−21.066282081426468270728867398472, −20.22343154501601689536129677347, −19.51131737500405113203365595957, −18.989462585228930163191299997228, −18.344994698126191159880579432138, −17.1474404580248739327097333988, −16.78873840037796637847179564866, −15.89450628422444827804595157670, −14.70996077292483253643042465059, −13.82038813061225890128330011751, −13.3681414487279165998401532128, −12.48023766769978974083055878042, −11.62760156681784980027892358097, −11.1027287380198123038930620303, −9.97473972658307018047759767509, −9.58894502666391493863663007383, −8.63709646020267018895701261967, −7.90199207222382125011580408620, −6.550085731302085891099282463514, −6.02325125897151710853521316818, −4.55016546898034438067687681645, −3.86023453624702865653672590402, −3.20268019756680508411078484664, −2.00889891944505565587447782277, −1.07851418654060361260461527182, 0.356782833845925009300102919711, 1.79151099764796963583260529205, 3.20928168386562959209694122616, 4.057601393854705656783458621522, 5.01130613731924353498154273623, 6.08742765506418446838770673355, 6.48101088712402419573074225435, 7.377921662716530513875993772737, 8.44461882855025023431280037939, 8.96915882425127331003694280737, 9.89286990742307280180680403437, 10.493997997023017485263168527076, 11.81860590527264579155351620746, 12.73791176040775783726112325826, 13.25512291938481073511836153429, 14.21210007355640620370571151253, 15.09671330447184541229798734536, 15.50397756336146046613723996053, 16.47770020593289936492121252617, 16.94939275600107454643912449311, 17.985760330821715363340319490246, 18.40678742809026264638849302519, 19.503212623460937111088645479408, 19.859070490057748239691303430060, 21.08482529671299879471551254182

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