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Properties

Label 1-2624-2624.125-r0-0-0
Degree $1$
Conductor $2624$
Sign $0.353 - 0.935i$
Analytic cond. $12.1858$
Root an. cond. $12.1858$
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  + (−0.382 + 0.923i)3-s + (−0.996 + 0.0784i)5-s + (0.156 + 0.987i)7-s + (−0.707 − 0.707i)9-s + (0.760 − 0.649i)11-s + (0.972 − 0.233i)13-s + (0.309 − 0.951i)15-s + (−0.309 − 0.951i)17-s + (0.522 + 0.852i)19-s + (−0.972 − 0.233i)21-s + (0.987 + 0.156i)23-s + (0.987 − 0.156i)25-s + (0.923 − 0.382i)27-s + (−0.760 − 0.649i)29-s + (−0.309 − 0.951i)31-s + ⋯
L(s)  = 1  + (−0.382 + 0.923i)3-s + (−0.996 + 0.0784i)5-s + (0.156 + 0.987i)7-s + (−0.707 − 0.707i)9-s + (0.760 − 0.649i)11-s + (0.972 − 0.233i)13-s + (0.309 − 0.951i)15-s + (−0.309 − 0.951i)17-s + (0.522 + 0.852i)19-s + (−0.972 − 0.233i)21-s + (0.987 + 0.156i)23-s + (0.987 − 0.156i)25-s + (0.923 − 0.382i)27-s + (−0.760 − 0.649i)29-s + (−0.309 − 0.951i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2624 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.353 - 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.353 - 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2624\)    =    \(2^{6} \cdot 41\)
Sign: $0.353 - 0.935i$
Analytic conductor: \(12.1858\)
Root analytic conductor: \(12.1858\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2624} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2624,\ (0:\ ),\ 0.353 - 0.935i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4799470691 - 0.3318345029i\)
\(L(\frac12)\) \(\approx\) \(0.4799470691 - 0.3318345029i\)
\(L(1)\) \(\approx\) \(0.7358277724 + 0.1646838141i\)
\(L(1)\) \(\approx\) \(0.7358277724 + 0.1646838141i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
41 \( 1 \)
good3 \( 1 + (-0.382 + 0.923i)T \)
5 \( 1 + (-0.996 + 0.0784i)T \)
7 \( 1 + (0.156 + 0.987i)T \)
11 \( 1 + (0.760 - 0.649i)T \)
13 \( 1 + (0.972 - 0.233i)T \)
17 \( 1 + (-0.309 - 0.951i)T \)
19 \( 1 + (0.522 + 0.852i)T \)
23 \( 1 + (0.987 + 0.156i)T \)
29 \( 1 + (-0.760 - 0.649i)T \)
31 \( 1 + (-0.309 - 0.951i)T \)
37 \( 1 + (-0.996 + 0.0784i)T \)
43 \( 1 + (-0.522 + 0.852i)T \)
47 \( 1 + (0.809 - 0.587i)T \)
53 \( 1 + (-0.996 + 0.0784i)T \)
59 \( 1 + (-0.852 - 0.522i)T \)
61 \( 1 + (-0.522 - 0.852i)T \)
67 \( 1 + (-0.760 - 0.649i)T \)
71 \( 1 + (0.453 - 0.891i)T \)
73 \( 1 + (-0.707 - 0.707i)T \)
79 \( 1 - T \)
83 \( 1 + (-0.382 + 0.923i)T \)
89 \( 1 + (-0.156 - 0.987i)T \)
97 \( 1 + (-0.951 - 0.309i)T \)
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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

−19.36416744549558830834645271187, −18.95799637581632418376829398454, −17.99671054059654135527059190249, −17.37401513937699245042428142475, −16.804034256856891008789001607102, −16.070676632667898949231790908525, −15.21988369536292939146053089983, −14.42428781701119867278965718852, −13.70541208406479511575240816672, −12.95462002813381191622048900884, −12.386930138363088567388248367634, −11.54803866687704957461030914024, −10.991429395197266180535083868616, −10.47876284954253918393267211514, −8.99838381756217260454737879662, −8.58011514003204540628434891105, −7.53971977508465903247588717260, −7.0430548658497224980055354382, −6.59783104758106817864268449895, −5.42573835507026962637451466701, −4.52955671088964620335824723419, −3.83385313648178263646174015450, −2.98929882394305409667629900087, −1.507796370640435543245432894372, −1.1771637299408251981682250561, 0.22169692205662455512201522480, 1.46985800420220375902308357781, 2.99025923013028871680794163161, 3.42741399142156869009346700701, 4.24905388276288288180213602722, 5.09901498760024184500476775660, 5.83536156778457383057795896441, 6.50906054316769456035792274277, 7.640505764498766540883469245878, 8.46426078482282987690708492770, 9.070613004514717456435093749735, 9.65075483898048528759918206986, 10.905236087209551444900302578541, 11.265633654837642816252986478623, 11.81735370313696380389995650969, 12.49165370513384680732397889342, 13.64304954092068321982807029240, 14.439289578668699851688576584398, 15.29247577634470837816139459081, 15.51652012492642388485044051, 16.369830497880046597580432564691, 16.79810793650401882811317749960, 17.83976192218077283628220118012, 18.65518787892350286909288195780, 19.011101960742910485233601257191

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