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Properties

Label 1-2624-2624.1299-r0-0-0
Degree $1$
Conductor $2624$
Sign $-0.879 + 0.475i$
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  + (0.923 − 0.382i)3-s + (−0.996 + 0.0784i)5-s + (−0.809 − 0.587i)7-s + (0.707 − 0.707i)9-s + (−0.996 − 0.0784i)11-s + (0.852 + 0.522i)13-s + (−0.891 + 0.453i)15-s + (−0.453 + 0.891i)17-s + (0.972 + 0.233i)19-s + (−0.972 − 0.233i)21-s + (−0.987 − 0.156i)23-s + (0.987 − 0.156i)25-s + (0.382 − 0.923i)27-s + (−0.0784 − 0.996i)29-s + (−0.309 − 0.951i)31-s + ⋯
L(s)  = 1  + (0.923 − 0.382i)3-s + (−0.996 + 0.0784i)5-s + (−0.809 − 0.587i)7-s + (0.707 − 0.707i)9-s + (−0.996 − 0.0784i)11-s + (0.852 + 0.522i)13-s + (−0.891 + 0.453i)15-s + (−0.453 + 0.891i)17-s + (0.972 + 0.233i)19-s + (−0.972 − 0.233i)21-s + (−0.987 − 0.156i)23-s + (0.987 − 0.156i)25-s + (0.382 − 0.923i)27-s + (−0.0784 − 0.996i)29-s + (−0.309 − 0.951i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2624 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.879 + 0.475i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.879 + 0.475i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2624\)    =    \(2^{6} \cdot 41\)
Sign: $-0.879 + 0.475i$
Analytic conductor: \(12.1858\)
Root analytic conductor: \(12.1858\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2624} (1299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2624,\ (0:\ ),\ -0.879 + 0.475i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02901434334 - 0.1145926808i\)
\(L(\frac12)\) \(\approx\) \(0.02901434334 - 0.1145926808i\)
\(L(1)\) \(\approx\) \(0.8772632187 - 0.1830765253i\)
\(L(1)\) \(\approx\) \(0.8772632187 - 0.1830765253i\)

Euler product

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

−20.00317277132137096550478125606, −19.087637240458661590116588209467, −18.39040895953873775797222831352, −18.07243689186281476061136886234, −16.39713059046023884628448157358, −16.08800162015323724836284140152, −15.45823908357400160574152982851, −15.18258710136443071023152860762, −13.89633959042406493369544953521, −13.517292824551712868717117536892, −12.59633837296357917856193379920, −12.04646171087278011304555125557, −10.98166537400766982551256890953, −10.38565247852507488801831181725, −9.48977692813785563196882416989, −8.82825508285275864723613978438, −8.19579857083383888821578050144, −7.47765169470004739159090326603, −6.783160396177118740109882784176, −5.52287402687363424312086990793, −4.884772427808264935758267528094, −3.88648881682806469806231562928, −3.07964191046750652324421705325, −2.815410059686157594429102702820, −1.46327052429339360032974071874, 0.03355318565360595917997747085, 1.227591410765344188014989608563, 2.335729137964171915689421457130, 3.18932627715132920042613675745, 3.907394311377946715472963052901, 4.32254837743174132644082526970, 5.853205214237030192805184379346, 6.5382339422427020246354900897, 7.50178002287794847587618412435, 7.84045976074671728777562941920, 8.607358392274556002610953038077, 9.45197147344346602491273554684, 10.219720976006670588883159654413, 10.99508495580396625583062146386, 11.85323637290594304807038587558, 12.70277383414466209513233772996, 13.24122567321979197123702606912, 13.86198319822831365369867440055, 14.64632743383907306449581465506, 15.60326006242831310824871465064, 15.85085875851266827374876431444, 16.583503304871020543044903828042, 17.718980039945355614129633799402, 18.593790670084462547623688169440, 18.88799147062359545448292776957

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