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Properties

Label 1-2624-2624.1229-r0-0-0
Degree $1$
Conductor $2624$
Sign $0.995 - 0.0980i$
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 + (0.923 − 0.382i)5-s + (0.707 − 0.707i)7-s + (−0.707 − 0.707i)9-s + (0.382 + 0.923i)11-s + (−0.923 − 0.382i)13-s − i·15-s − i·17-s + (0.923 + 0.382i)19-s + (0.382 + 0.923i)21-s + (0.707 + 0.707i)23-s + (0.707 − 0.707i)25-s + (0.923 − 0.382i)27-s + (0.382 − 0.923i)29-s − 31-s + ⋯
L(s)  = 1  + (−0.382 + 0.923i)3-s + (0.923 − 0.382i)5-s + (0.707 − 0.707i)7-s + (−0.707 − 0.707i)9-s + (0.382 + 0.923i)11-s + (−0.923 − 0.382i)13-s − i·15-s − i·17-s + (0.923 + 0.382i)19-s + (0.382 + 0.923i)21-s + (0.707 + 0.707i)23-s + (0.707 − 0.707i)25-s + (0.923 − 0.382i)27-s + (0.382 − 0.923i)29-s − 31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2624 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2624\)    =    \(2^{6} \cdot 41\)
Sign: $0.995 - 0.0980i$
Analytic conductor: \(12.1858\)
Root analytic conductor: \(12.1858\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2624} (1229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2624,\ (0:\ ),\ 0.995 - 0.0980i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.895219801 - 0.09310617847i\)
\(L(\frac12)\) \(\approx\) \(1.895219801 - 0.09310617847i\)
\(L(1)\) \(\approx\) \(1.204854918 + 0.1007116971i\)
\(L(1)\) \(\approx\) \(1.204854918 + 0.1007116971i\)

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 \( 1 + (0.923 - 0.382i)T \)
7 \( 1 + (0.707 - 0.707i)T \)
11 \( 1 + (0.382 + 0.923i)T \)
13 \( 1 + (-0.923 - 0.382i)T \)
17 \( 1 - iT \)
19 \( 1 + (0.923 + 0.382i)T \)
23 \( 1 + (0.707 + 0.707i)T \)
29 \( 1 + (0.382 - 0.923i)T \)
31 \( 1 - T \)
37 \( 1 + (-0.923 + 0.382i)T \)
43 \( 1 + (0.382 + 0.923i)T \)
47 \( 1 - iT \)
53 \( 1 + (0.382 + 0.923i)T \)
59 \( 1 + (0.923 - 0.382i)T \)
61 \( 1 + (0.382 - 0.923i)T \)
67 \( 1 + (-0.382 + 0.923i)T \)
71 \( 1 + (0.707 - 0.707i)T \)
73 \( 1 + (-0.707 - 0.707i)T \)
79 \( 1 - iT \)
83 \( 1 + (-0.923 - 0.382i)T \)
89 \( 1 + (-0.707 + 0.707i)T \)
97 \( 1 + T \)
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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

−19.16562359227318208640416206742, −18.58578890859950418125707668698, −17.968021126751301419854001521689, −17.33546301945841096586788894140, −16.839749553280061092318662446777, −15.957170330491687760569471902495, −14.65605367700242518754895246540, −14.44744460044225823997480221174, −13.7151438745197068543672673141, −12.84359482615966075181882011603, −12.28786888357943970572139972975, −11.42126370787921703967829036175, −10.92454561213840859366957973560, −10.083630790777853655990745045148, −8.81660112727232587152247593297, −8.729715967215877305003369554639, −7.42392911207047394683401120239, −6.91562945895903495041149561799, −6.009885561522159343075599904527, −5.485404012414369700508579491291, −4.82377907443329511442089333531, −3.34469514889351170808700502136, −2.47632302077176812950572922087, −1.832911793376638657750220722212, −1.00577785664388087607583585368, 0.734089267129276951441557664044, 1.71445457037925634939981663173, 2.737060134210127768654328581, 3.7481130949371531803851797859, 4.74486067112975474819486568579, 5.04598435407761425983195936281, 5.77189294442207411192520483206, 6.935590245879666976792153371568, 7.50413371310170481345502785482, 8.62602988532508199190424226726, 9.56291089885197289008724928156, 9.79990553228523491579107516121, 10.51233563007825088149701767302, 11.46269977986055246473203673226, 11.998021007886330719997587088701, 12.902820711036462949400260322971, 13.86085652990445229109538238362, 14.35410804359724469837932888201, 15.05201393595542302436110987919, 15.84236738003376316336040169993, 16.74555772145484815043159102483, 17.15976949307010920268820033621, 17.72641718017538460594224188227, 18.25265235311936336200063451885, 19.66916554605168812931071910040

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