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Properties

Label 1-4275-4275.2612-r0-0-0
Degree $1$
Conductor $4275$
Sign $-0.246 - 0.969i$
Analytic cond. $19.8530$
Root an. cond. $19.8530$
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  + (0.927 + 0.374i)2-s + (0.719 + 0.694i)4-s + (−0.866 − 0.5i)7-s + (0.406 + 0.913i)8-s + (−0.309 − 0.951i)11-s + (0.788 + 0.615i)13-s + (−0.615 − 0.788i)14-s + (0.0348 + 0.999i)16-s + (0.275 − 0.961i)17-s + (0.0697 − 0.997i)22-s + (−0.469 − 0.882i)23-s + (0.5 + 0.866i)26-s + (−0.275 − 0.961i)28-s + (−0.719 − 0.694i)29-s + (−0.809 − 0.587i)31-s + (−0.342 + 0.939i)32-s + ⋯
L(s)  = 1  + (0.927 + 0.374i)2-s + (0.719 + 0.694i)4-s + (−0.866 − 0.5i)7-s + (0.406 + 0.913i)8-s + (−0.309 − 0.951i)11-s + (0.788 + 0.615i)13-s + (−0.615 − 0.788i)14-s + (0.0348 + 0.999i)16-s + (0.275 − 0.961i)17-s + (0.0697 − 0.997i)22-s + (−0.469 − 0.882i)23-s + (0.5 + 0.866i)26-s + (−0.275 − 0.961i)28-s + (−0.719 − 0.694i)29-s + (−0.809 − 0.587i)31-s + (−0.342 + 0.939i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.246 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.246 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4275\)    =    \(3^{2} \cdot 5^{2} \cdot 19\)
Sign: $-0.246 - 0.969i$
Analytic conductor: \(19.8530\)
Root analytic conductor: \(19.8530\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4275} (2612, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4275,\ (0:\ ),\ -0.246 - 0.969i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8005677922 - 1.030124287i\)
\(L(\frac12)\) \(\approx\) \(0.8005677922 - 1.030124287i\)
\(L(1)\) \(\approx\) \(1.415181195 + 0.08587064490i\)
\(L(1)\) \(\approx\) \(1.415181195 + 0.08587064490i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.927 + 0.374i)T \)
7 \( 1 + (-0.866 - 0.5i)T \)
11 \( 1 + (-0.309 - 0.951i)T \)
13 \( 1 + (0.788 + 0.615i)T \)
17 \( 1 + (0.275 - 0.961i)T \)
23 \( 1 + (-0.469 - 0.882i)T \)
29 \( 1 + (-0.719 - 0.694i)T \)
31 \( 1 + (-0.809 - 0.587i)T \)
37 \( 1 + (0.951 + 0.309i)T \)
41 \( 1 + (-0.848 + 0.529i)T \)
43 \( 1 + (-0.984 - 0.173i)T \)
47 \( 1 + (-0.694 + 0.719i)T \)
53 \( 1 + (-0.970 - 0.241i)T \)
59 \( 1 + (-0.882 - 0.469i)T \)
61 \( 1 + (0.0348 - 0.999i)T \)
67 \( 1 + (-0.0697 - 0.997i)T \)
71 \( 1 + (0.997 + 0.0697i)T \)
73 \( 1 + (-0.139 - 0.990i)T \)
79 \( 1 + (-0.559 + 0.829i)T \)
83 \( 1 + (0.406 + 0.913i)T \)
89 \( 1 + (-0.882 + 0.469i)T \)
97 \( 1 + (0.0697 - 0.997i)T \)
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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

−18.57537773051835950657703560718, −18.1165898162220420477049169092, −17.103721199691603042482620179583, −16.21012191521282937267334843785, −15.761669013479516177983154297000, −14.99870406391564151099347534743, −14.68452878997515515664261489384, −13.51863434478611254476260609580, −13.07923135521603759943380400036, −12.55152192865176051979392667776, −11.94620313232347293671322313057, −11.0897343373632935318480465387, −10.36736224767987783544357494100, −9.84916448549083177889132929885, −9.06273932022146096285765174096, −8.07520776632427984857294515282, −7.20580296265960967213779240544, −6.53273172490046214906720472638, −5.68524128463623098362229238886, −5.36971568217949631949472951410, −4.26180380269211574496778962830, −3.527765175557047847885978066109, −3.02985270687734901154242303098, −1.9786679543432466144013110854, −1.3923813769598551509087960577, 0.22667897190630249245037946775, 1.55802520277621980088919917700, 2.63174660857872477293499796680, 3.319870185359140310607683698883, 3.89159262294788157979461027951, 4.716747452006203675581696181570, 5.55402832291303966995055229804, 6.4232345063887529595390113718, 6.57813816424399261283107529197, 7.72965733770296185818685810881, 8.16606679183172383380009869087, 9.21384956027520593848974320746, 9.88335000771831228195983418751, 11.02785385979509426881401889770, 11.233296175221437864879199654512, 12.18281315827244177422028479203, 12.92658976438929800049599528797, 13.54140636217817643594985418187, 13.897641613787504383741314836556, 14.67752933295689770208260269370, 15.553930847251866720829999588779, 16.15756756195760731147211556212, 16.597571656166959430266751364575, 17.04702880496771836280945662011, 18.422146799884634381894387388663

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