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Properties

Label 1-2624-2624.1203-r0-0-0
Degree $1$
Conductor $2624$
Sign $0.999 + 0.0230i$
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 + (−0.382 − 0.923i)5-s − i·7-s + (−0.707 + 0.707i)9-s + (0.923 + 0.382i)11-s + (−0.923 + 0.382i)13-s + (−0.707 + 0.707i)15-s + (0.707 − 0.707i)17-s + (−0.382 − 0.923i)19-s + (0.923 − 0.382i)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 + (−0.382 − 0.923i)5-s − i·7-s + (−0.707 + 0.707i)9-s + (0.923 + 0.382i)11-s + (−0.923 + 0.382i)13-s + (−0.707 + 0.707i)15-s + (0.707 − 0.707i)17-s + (−0.382 − 0.923i)19-s + (0.923 − 0.382i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 2624 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0230i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0230i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2624\)    =    \(2^{6} \cdot 41\)
Sign: $0.999 + 0.0230i$
Analytic conductor: \(12.1858\)
Root analytic conductor: \(12.1858\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2624} (1203, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2624,\ (0:\ ),\ 0.999 + 0.0230i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.003833863 + 0.01157064357i\)
\(L(\frac12)\) \(\approx\) \(1.003833863 + 0.01157064357i\)
\(L(1)\) \(\approx\) \(0.8054750187 - 0.1978559898i\)
\(L(1)\) \(\approx\) \(0.8054750187 - 0.1978559898i\)

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

−19.42091701911026823267210630501, −18.75033794391840997742224116614, −17.74010529673244119261961313029, −17.03252929050559104549568900410, −16.66524046617337277070043171729, −15.90173887595656253236788690488, −14.79893312721243330752493115458, −14.60257008907875617565078190479, −14.07916513090718168315901276724, −12.69214831372553890804073467330, −12.15200675626322652189669642123, −11.0942452498554768000490464766, −10.77726889139599241913803912993, −10.125793701397650750029853143658, −9.43951736605662583493612327792, −8.434815135807601063759876249165, −7.579374614217638357969224065201, −6.80992059309958372081148685439, −6.13508293596741282885887071253, −5.20876305487454470990301551975, −4.28648468798478717958708859575, −3.56226319042684331779341749512, −3.21234987604241254504357969297, −1.774241622683865383234808013199, −0.45523604219162715267061723313, 0.7917781198271032915069482874, 1.75411244118729390307094342779, 2.44304298573831561939433167237, 3.566423284450570353560728025028, 4.81572766735270088703415185840, 5.16837368830605970899789910112, 6.06671335736052424837035574010, 6.97601334158604915201631856037, 7.55295769690918339647041555044, 8.401835509372241343854553626259, 9.260309337228810122444993352367, 9.544280234821647838198250419512, 11.14856264706725088207905748026, 11.61766274721074153385672392578, 12.29071601802532861544482938678, 12.62819693260107675334718907459, 13.49922192682102259192671523031, 14.34013706548834061337817044806, 15.096898555678467117544977695789, 15.870313261348770848608868682358, 16.73690576856063634385195106678, 17.241804112468920270409905574079, 17.79927196195375465646656981019, 18.85102221375956857286037162810, 19.27574556415172271509122572946

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