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Properties

Label 1-2624-2624.1333-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: $\chi_{2624} (1333, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2624,\ (0:\ ),\ 0.353 + 0.935i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.669840193 + 1.154524375i\)
\(L(\frac12)\) \(\approx\) \(1.669840193 + 1.154524375i\)
\(L(1)\) \(\approx\) \(1.265705306 + 0.3736177351i\)
\(L(1)\) \(\approx\) \(1.265705306 + 0.3736177351i\)

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

−19.04521549957446062498109656255, −18.461213405710349831907962758921, −17.84414990905522513579933684562, −17.417324568676564225836647402099, −16.53911358320452167352211722013, −15.41415138312721547104588069574, −14.87163283081601020051204417093, −14.23851700231449044396514989471, −13.25510517752318875446714319477, −13.00646295377279846544492016061, −12.16434230034546783836754194686, −11.53420483337920277472153935960, −10.509213223071888061264737540341, −9.532882314336360831225122200664, −9.07311258441200450695795720053, −8.40800374358400098554985205740, −7.20613904469654352112689942087, −7.00275416751643717222457359785, −5.86874767247747741875142890312, −5.2736634781526693366248995595, −4.5291902703512084736238785608, −2.81194856271011313865424568574, −2.50077187614052097618672352703, −1.93116789922631686377322674389, −0.6639256360703039593141472842, 0.99268533430353463376911697470, 2.23431024866188715874098142848, 2.848958294229735057662729777392, 3.81036158368242072435066347202, 4.61331816538627617289301794591, 5.32189569405860349407931513225, 6.08030627438200309516673887893, 7.02780786782836458805778202597, 8.04382232078173888600047953199, 8.53580037552170073206473878610, 9.59503022332942613842957791138, 10.10000860665122196201312727011, 10.69967175816430440023093359843, 11.14874412206448693198938930002, 12.61281285468196602992641591030, 13.14266144908788776825837353291, 13.98575598385940345514978309988, 14.452925818416191583119227112300, 15.09002707529432021713474230487, 16.03102944569692627108919880837, 16.776784191349758919457654317378, 17.185645520761905298308421359295, 17.84722266078063838463305572145, 18.96330870637463447315262068850, 19.53954257319150733128087442326

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