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Properties

Label 1-2624-2624.1021-r0-0-0
Degree $1$
Conductor $2624$
Sign $0.283 + 0.959i$
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 + (0.233 − 0.972i)5-s + (0.891 + 0.453i)7-s + (0.707 + 0.707i)9-s + (−0.522 + 0.852i)11-s + (−0.760 − 0.649i)13-s + (−0.587 + 0.809i)15-s + (−0.587 − 0.809i)17-s + (−0.996 − 0.0784i)19-s + (−0.649 − 0.760i)21-s + (−0.891 + 0.453i)23-s + (−0.891 − 0.453i)25-s + (−0.382 − 0.923i)27-s + (−0.522 − 0.852i)29-s + (0.809 − 0.587i)31-s + ⋯
L(s)  = 1  + (−0.923 − 0.382i)3-s + (0.233 − 0.972i)5-s + (0.891 + 0.453i)7-s + (0.707 + 0.707i)9-s + (−0.522 + 0.852i)11-s + (−0.760 − 0.649i)13-s + (−0.587 + 0.809i)15-s + (−0.587 − 0.809i)17-s + (−0.996 − 0.0784i)19-s + (−0.649 − 0.760i)21-s + (−0.891 + 0.453i)23-s + (−0.891 − 0.453i)25-s + (−0.382 − 0.923i)27-s + (−0.522 − 0.852i)29-s + (0.809 − 0.587i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2624 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.283 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.283 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2624\)    =    \(2^{6} \cdot 41\)
Sign: $0.283 + 0.959i$
Analytic conductor: \(12.1858\)
Root analytic conductor: \(12.1858\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2624} (1021, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2624,\ (0:\ ),\ 0.283 + 0.959i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4149393021 + 0.3101626244i\)
\(L(\frac12)\) \(\approx\) \(0.4149393021 + 0.3101626244i\)
\(L(1)\) \(\approx\) \(0.7078091370 - 0.1134336739i\)
\(L(1)\) \(\approx\) \(0.7078091370 - 0.1134336739i\)

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 \( 1 + (0.233 - 0.972i)T \)
7 \( 1 + (0.891 + 0.453i)T \)
11 \( 1 + (-0.522 + 0.852i)T \)
13 \( 1 + (-0.760 - 0.649i)T \)
17 \( 1 + (-0.587 - 0.809i)T \)
19 \( 1 + (-0.996 - 0.0784i)T \)
23 \( 1 + (-0.891 + 0.453i)T \)
29 \( 1 + (-0.522 - 0.852i)T \)
31 \( 1 + (0.809 - 0.587i)T \)
37 \( 1 + (-0.233 + 0.972i)T \)
43 \( 1 + (0.0784 + 0.996i)T \)
47 \( 1 + (0.951 + 0.309i)T \)
53 \( 1 + (-0.972 - 0.233i)T \)
59 \( 1 + (0.996 - 0.0784i)T \)
61 \( 1 + (0.0784 - 0.996i)T \)
67 \( 1 + (0.522 + 0.852i)T \)
71 \( 1 + (-0.156 + 0.987i)T \)
73 \( 1 + (0.707 + 0.707i)T \)
79 \( 1 - iT \)
83 \( 1 + (-0.382 + 0.923i)T \)
89 \( 1 + (-0.891 - 0.453i)T \)
97 \( 1 + (0.809 + 0.587i)T \)
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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

−19.02814290300317164049057168413, −18.3869337202959345785939066746, −17.69668749102124801952997628014, −17.17139210896193108538348209767, −16.52870526989555459611842766813, −15.648556710972444156706647923470, −14.94072995676644216641050501245, −14.30794289662510727365512440222, −13.654625214817777677839577248102, −12.62075488184358685544648964546, −11.882844612185537834068299156094, −11.05079606652101558148860323954, −10.63596748886631733908413429657, −10.23262039036796904585484607898, −9.101492066425530191687247756182, −8.23841782163394614121491385153, −7.29348342378118146931662642443, −6.64388052894154521030748067701, −5.93307000913494431442358663369, −5.15891268377886606000465930170, −4.2631508057196424018585017486, −3.68622016831844677173885426821, −2.398619735766360971782319754, −1.668003647571212602989563625035, −0.20948031332446579421198790093, 0.93258999270605006176655158031, 2.02173791627315901716936352881, 2.40851075848311001734405638681, 4.32896994690181282425530567621, 4.69562465566643673472435686173, 5.39033735373420968138818877422, 6.03041842786645434307995051350, 7.05947963048601794802977372531, 7.918875364291139662463257683775, 8.30870208643231922709337002932, 9.592686549133590763527497239995, 9.99159087262404244129401341598, 11.05590693818143040343893734424, 11.71736714616768259407512059042, 12.31171997597132345707653180300, 12.93894863442064228106979375192, 13.530387406558153098824558574774, 14.55440175128265099601021716363, 15.569452755637260633708433928555, 15.78290307265827724466074344509, 17.09267285408305575875742367687, 17.29610146319229472915677344071, 17.87654763543602963267834474497, 18.595565117211197142371567856895, 19.449654198284923403567222085252

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