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Properties

Label 1-2624-2624.1261-r0-0-0
Degree $1$
Conductor $2624$
Sign $0.827 + 0.562i$
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 + (0.522 − 0.852i)5-s + (−0.453 − 0.891i)7-s + (−0.707 − 0.707i)9-s + (−0.233 + 0.972i)11-s + (−0.0784 + 0.996i)13-s + (−0.587 − 0.809i)15-s + (−0.587 + 0.809i)17-s + (−0.649 + 0.760i)19-s + (−0.996 + 0.0784i)21-s + (−0.453 + 0.891i)23-s + (−0.453 − 0.891i)25-s + (−0.923 + 0.382i)27-s + (−0.233 − 0.972i)29-s + (0.809 + 0.587i)31-s + ⋯
L(s)  = 1  + (0.382 − 0.923i)3-s + (0.522 − 0.852i)5-s + (−0.453 − 0.891i)7-s + (−0.707 − 0.707i)9-s + (−0.233 + 0.972i)11-s + (−0.0784 + 0.996i)13-s + (−0.587 − 0.809i)15-s + (−0.587 + 0.809i)17-s + (−0.649 + 0.760i)19-s + (−0.996 + 0.0784i)21-s + (−0.453 + 0.891i)23-s + (−0.453 − 0.891i)25-s + (−0.923 + 0.382i)27-s + (−0.233 − 0.972i)29-s + (0.809 + 0.587i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2624 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.827 + 0.562i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.827 + 0.562i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2624\)    =    \(2^{6} \cdot 41\)
Sign: $0.827 + 0.562i$
Analytic conductor: \(12.1858\)
Root analytic conductor: \(12.1858\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2624} (1261, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2624,\ (0:\ ),\ 0.827 + 0.562i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9674825423 + 0.2976730383i\)
\(L(\frac12)\) \(\approx\) \(0.9674825423 + 0.2976730383i\)
\(L(1)\) \(\approx\) \(0.9670504062 - 0.3018967282i\)
\(L(1)\) \(\approx\) \(0.9670504062 - 0.3018967282i\)

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

−19.26056336911718174694475391735, −18.632171770343603882430104873655, −17.90859436702095440188301235200, −17.19826133943573203124087932615, −16.1182023972026557438787496243, −15.80206382469647587221924310492, −14.99685880267264744000800620401, −14.49653851244047375130553302125, −13.62392198055720698032461043968, −13.092263627303427182645412744352, −12.02704036594861432608052740177, −11.01766514314103226149108222778, −10.693123257366386547746336788121, −9.90245013756540678429690158218, −9.09024679410797760579309432355, −8.64490203891828131958363107962, −7.69221170370862642685454933600, −6.635813006535179955181186233006, −5.85852252823974653748426352350, −5.33571011572934603761688381524, −4.32798152032311639775864563509, −3.23710595229654158465408564637, −2.75436443372444091122006185628, −2.227343603495607423514127731995, −0.294895295090874638628779744376, 1.13015433250209929851966058987, 1.81604588636071586640720066173, 2.47498843300350165106039013194, 3.910511798685188634156228686143, 4.28070430777778426283539385423, 5.49758936388420616441762472337, 6.34268927167145659640261735938, 6.90409532135906598765791604486, 7.7415102687546677079795179325, 8.44318599439944941778027504862, 9.24025153635082215708341120478, 9.90458896702266989376554023808, 10.639453055171779719368270080499, 11.95253146361480542616395338314, 12.25615905864053064851667433654, 13.19794848391367340338458013229, 13.517431238494111011212965024760, 14.1884487270667369357900231974, 15.08282497540325378362951737486, 15.92591006969254203185084998969, 16.89423902817390729578996446324, 17.327715735220587611684862836752, 17.78495604542244136580910504128, 18.9704530217311366340936230854, 19.32652992377089192377447591434

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