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Properties

Label 1-1035-1035.302-r0-0-0
Degree $1$
Conductor $1035$
Sign $0.561 - 0.827i$
Analytic cond. $4.80651$
Root an. cond. $4.80651$
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  + (−0.971 − 0.235i)2-s + (0.888 + 0.458i)4-s + (−0.814 + 0.580i)7-s + (−0.755 − 0.654i)8-s + (−0.723 + 0.690i)11-s + (−0.814 − 0.580i)13-s + (0.928 − 0.371i)14-s + (0.580 + 0.814i)16-s + (0.540 − 0.841i)17-s + (−0.841 + 0.540i)19-s + (0.866 − 0.5i)22-s + (0.654 + 0.755i)26-s + (−0.989 + 0.142i)28-s + (−0.888 + 0.458i)29-s + (0.981 + 0.189i)31-s + (−0.371 − 0.928i)32-s + ⋯
L(s)  = 1  + (−0.971 − 0.235i)2-s + (0.888 + 0.458i)4-s + (−0.814 + 0.580i)7-s + (−0.755 − 0.654i)8-s + (−0.723 + 0.690i)11-s + (−0.814 − 0.580i)13-s + (0.928 − 0.371i)14-s + (0.580 + 0.814i)16-s + (0.540 − 0.841i)17-s + (−0.841 + 0.540i)19-s + (0.866 − 0.5i)22-s + (0.654 + 0.755i)26-s + (−0.989 + 0.142i)28-s + (−0.888 + 0.458i)29-s + (0.981 + 0.189i)31-s + (−0.371 − 0.928i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1035 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.561 - 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1035 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.561 - 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1035\)    =    \(3^{2} \cdot 5 \cdot 23\)
Sign: $0.561 - 0.827i$
Analytic conductor: \(4.80651\)
Root analytic conductor: \(4.80651\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1035} (302, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1035,\ (0:\ ),\ 0.561 - 0.827i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4691939278 - 0.2486571655i\)
\(L(\frac12)\) \(\approx\) \(0.4691939278 - 0.2486571655i\)
\(L(1)\) \(\approx\) \(0.5552758096 - 0.04309033007i\)
\(L(1)\) \(\approx\) \(0.5552758096 - 0.04309033007i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.971 - 0.235i)T \)
7 \( 1 + (-0.814 + 0.580i)T \)
11 \( 1 + (-0.723 + 0.690i)T \)
13 \( 1 + (-0.814 - 0.580i)T \)
17 \( 1 + (0.540 - 0.841i)T \)
19 \( 1 + (-0.841 + 0.540i)T \)
29 \( 1 + (-0.888 + 0.458i)T \)
31 \( 1 + (0.981 + 0.189i)T \)
37 \( 1 + (-0.989 - 0.142i)T \)
41 \( 1 + (0.786 - 0.618i)T \)
43 \( 1 + (-0.189 - 0.981i)T \)
47 \( 1 + (0.866 + 0.5i)T \)
53 \( 1 + (0.909 + 0.415i)T \)
59 \( 1 + (0.580 - 0.814i)T \)
61 \( 1 + (-0.327 - 0.945i)T \)
67 \( 1 + (-0.690 + 0.723i)T \)
71 \( 1 + (0.959 - 0.281i)T \)
73 \( 1 + (-0.540 - 0.841i)T \)
79 \( 1 + (0.995 + 0.0950i)T \)
83 \( 1 + (0.618 - 0.786i)T \)
89 \( 1 + (-0.654 - 0.755i)T \)
97 \( 1 + (-0.371 + 0.928i)T \)
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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

−21.38030473272044147218308395338, −20.9358305632135107060542669640, −19.68913061979904486559212185029, −19.368017694408920825293118650822, −18.72407557859853149135672548655, −17.706729002917078248478486602107, −16.80843498849634504799156953566, −16.57728215522257707625387646028, −15.52837007495797341133392552126, −14.87582342971176246203546701836, −13.83126009881754284111490539587, −12.96299300496362061585248952270, −12.02720452537670875515283938075, −11.03700029663421275425057691424, −10.32567075474327663166131418274, −9.70027894803183039560414358672, −8.77322923475347815584841795663, −7.94113840177543364884122473695, −7.12569567270137113014371186429, −6.34174669285400247166386569548, −5.53133470293885915246825108612, −4.22032829925534778275280854160, −3.037259674232909917349715986116, −2.16346494043856449284822473850, −0.8007836090294806546302960191, 0.41380270473945557398887953348, 2.02637122362215599370166500134, 2.68604475947978726420188684593, 3.62152458478159744725433162458, 5.084873268311488159425296411934, 5.98090017782138563131914759237, 7.05774229174882862777889866981, 7.63935924498956462355479594643, 8.643716591121612251124275061596, 9.465204698133927012067251250630, 10.13992283428246978281085061172, 10.75025473550119215192080641051, 12.17842598150143742544851624002, 12.290875698177733999264768249965, 13.26920843596931713294298263701, 14.619876506889415398077591121184, 15.45069447522518407297259756944, 15.96884542926859627370002476505, 16.91959894966990402770111249433, 17.567744160361251158689776879, 18.50492414609562317293505767597, 18.970020999268651594481038522647, 19.77060413568898476808846770340, 20.60128759688454131167118712714, 21.16488697708964493620039005251

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