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Properties

Label 1-2624-2624.1197-r0-0-0
Degree $1$
Conductor $2624$
Sign $-0.696 - 0.717i$
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 + (−0.522 + 0.852i)5-s + (−0.891 + 0.453i)7-s + (0.707 + 0.707i)9-s + (0.972 + 0.233i)11-s + (0.996 + 0.0784i)13-s + (0.809 − 0.587i)15-s + (−0.809 − 0.587i)17-s + (−0.760 − 0.649i)19-s + (0.996 − 0.0784i)21-s + (−0.453 + 0.891i)23-s + (−0.453 − 0.891i)25-s + (−0.382 − 0.923i)27-s + (−0.972 + 0.233i)29-s + (0.809 + 0.587i)31-s + ⋯
L(s)  = 1  + (−0.923 − 0.382i)3-s + (−0.522 + 0.852i)5-s + (−0.891 + 0.453i)7-s + (0.707 + 0.707i)9-s + (0.972 + 0.233i)11-s + (0.996 + 0.0784i)13-s + (0.809 − 0.587i)15-s + (−0.809 − 0.587i)17-s + (−0.760 − 0.649i)19-s + (0.996 − 0.0784i)21-s + (−0.453 + 0.891i)23-s + (−0.453 − 0.891i)25-s + (−0.382 − 0.923i)27-s + (−0.972 + 0.233i)29-s + (0.809 + 0.587i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2624 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.696 - 0.717i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.696 - 0.717i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2624\)    =    \(2^{6} \cdot 41\)
Sign: $-0.696 - 0.717i$
Analytic conductor: \(12.1858\)
Root analytic conductor: \(12.1858\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2624} (1197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2624,\ (0:\ ),\ -0.696 - 0.717i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.004866655619 + 0.01150400809i\)
\(L(\frac12)\) \(\approx\) \(0.004866655619 + 0.01150400809i\)
\(L(1)\) \(\approx\) \(0.5812902703 + 0.08944462253i\)
\(L(1)\) \(\approx\) \(0.5812902703 + 0.08944462253i\)

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

−19.61673809968917406029711162089, −18.971560297503145686541330815548, −18.14488594139009596485263881226, −17.16328388795005925346065104150, −16.672638316185726836463966544276, −16.37565223501119954902140439462, −15.449259139210460556210090716425, −14.99308720003867007547053566052, −13.616944098462868198617003072, −13.17300348683696243772544081245, −12.18187564081974342117392847130, −11.98384763210583702865564890733, −10.76557501987685789180683023065, −10.55666962965620579185151675377, −9.35831667001610310420394980040, −8.91775743262238940722576083155, −8.02185433488102492465824610518, −6.92140958177143266648576130265, −6.22159479737573734434745591151, −5.78616980865295579520734357181, −4.538118376180014177190387730944, −3.96109915880398425895677592410, −3.593902232094623861995598794959, −1.87576587858923429981825837354, −0.898098866839323615189534961176, 0.005832941737393723885214506824, 1.35964870543019339575531740483, 2.34439064413533334684475557481, 3.36134652308380834817331413178, 4.08123315991850109933477081496, 5.02008880114826696329871371078, 6.15297028071588125533595319627, 6.534257043660983718379162949840, 7.007413289290622431951699472, 8.017500153001671656644964535624, 8.98665001720070524325630076365, 9.747568399756168798049417376058, 10.63299778266444919162420557890, 11.33483513751140073915078190605, 11.740969807347974426931570027250, 12.537300516043344558399900195941, 13.347273144268705867380414986, 13.92187508151340396698770957034, 15.085334450792865989979582972249, 15.622558204911279900591490456, 16.16723632510903333150622169391, 17.03290531819387213400935190564, 17.795506507612348493123806048344, 18.33779987669539457872900586906, 19.107973036598572985605934271749

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