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Properties

Label 1-2624-2624.1005-r0-0-0
Degree $1$
Conductor $2624$
Sign $-0.226 + 0.974i$
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 + (0.649 + 0.760i)5-s + (0.987 + 0.156i)7-s + (0.707 + 0.707i)9-s + (0.0784 + 0.996i)11-s + (−0.852 + 0.522i)13-s + (−0.309 − 0.951i)15-s + (0.309 − 0.951i)17-s + (0.233 + 0.972i)19-s + (−0.852 − 0.522i)21-s + (−0.156 − 0.987i)23-s + (−0.156 + 0.987i)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 + (0.649 + 0.760i)5-s + (0.987 + 0.156i)7-s + (0.707 + 0.707i)9-s + (0.0784 + 0.996i)11-s + (−0.852 + 0.522i)13-s + (−0.309 − 0.951i)15-s + (0.309 − 0.951i)17-s + (0.233 + 0.972i)19-s + (−0.852 − 0.522i)21-s + (−0.156 − 0.987i)23-s + (−0.156 + 0.987i)25-s + (−0.382 − 0.923i)27-s + (−0.0784 + 0.996i)29-s + (−0.309 + 0.951i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2624 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.226 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.226 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2624\)    =    \(2^{6} \cdot 41\)
Sign: $-0.226 + 0.974i$
Analytic conductor: \(12.1858\)
Root analytic conductor: \(12.1858\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2624} (1005, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2624,\ (0:\ ),\ -0.226 + 0.974i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8263338664 + 1.040144234i\)
\(L(\frac12)\) \(\approx\) \(0.8263338664 + 1.040144234i\)
\(L(1)\) \(\approx\) \(0.9379116105 + 0.2524803863i\)
\(L(1)\) \(\approx\) \(0.9379116105 + 0.2524803863i\)

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

−19.19434795586948440550483673952, −18.03396359464129140374606701285, −17.644807673002550707082119393261, −17.036987238783129363095556920132, −16.58795956499012797114412670516, −15.667672223815725653733399637433, −14.9899496381928496474729347118, −14.17199198657655207119843685918, −13.30099676331814885986638618800, −12.73155527919484618324013918851, −11.83131885482505821541706867581, −11.2342126639340689661946225920, −10.62615116158700519063842312421, −9.66926129001772944692445281754, −9.24063029734225185190718733619, −8.09930670635870759642918906411, −7.60028397110297261819611369493, −6.27072853563380092624512503606, −5.77851845141778803167962503590, −5.067917201597359259320678683209, −4.492456282288020964531038206604, −3.56445729034118931735486777052, −2.27518935787060482090598061450, −1.309797540151923632063035691008, −0.50545675314822094013686810049, 1.26938535726965218397156176656, 1.95878918270966587407807349032, 2.65999800620688197704301941043, 4.05202067256076152540382114952, 5.05986772590979245043338115346, 5.28668908910223680043345119425, 6.42878929539529214385275457675, 7.08019755555128112868748527785, 7.51982095250411632797519130832, 8.60135696529581571621686360606, 9.771204844186938161579452587065, 10.16149086198160536934821098699, 10.96296886657538425720431085484, 11.76096107466361833626795635874, 12.19629335035809773783321897233, 13.00517927132335106676469524760, 14.086155393850330616458306350644, 14.449576029994140362055036802106, 15.122863836873601617572719732566, 16.29428593047751438506967832242, 16.79705709491840160819802679855, 17.665678830433617763081513676386, 18.06124814988541293493759929437, 18.533673933458008114451599712847, 19.316645626153642823239782221479

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