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Properties

Label 1-2349-2349.1159-r0-0-0
Degree $1$
Conductor $2349$
Sign $0.466 + 0.884i$
Analytic cond. $10.9087$
Root an. cond. $10.9087$
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  + (−0.597 + 0.802i)2-s + (−0.286 − 0.957i)4-s + (−0.835 − 0.549i)5-s + (−0.686 − 0.727i)7-s + (0.939 + 0.342i)8-s + (0.939 − 0.342i)10-s + (−0.893 − 0.448i)11-s + (0.396 − 0.918i)13-s + (0.993 − 0.116i)14-s + (−0.835 + 0.549i)16-s + (−0.766 − 0.642i)17-s + (−0.766 + 0.642i)19-s + (−0.286 + 0.957i)20-s + (0.893 − 0.448i)22-s + (−0.686 + 0.727i)23-s + ⋯
L(s)  = 1  + (−0.597 + 0.802i)2-s + (−0.286 − 0.957i)4-s + (−0.835 − 0.549i)5-s + (−0.686 − 0.727i)7-s + (0.939 + 0.342i)8-s + (0.939 − 0.342i)10-s + (−0.893 − 0.448i)11-s + (0.396 − 0.918i)13-s + (0.993 − 0.116i)14-s + (−0.835 + 0.549i)16-s + (−0.766 − 0.642i)17-s + (−0.766 + 0.642i)19-s + (−0.286 + 0.957i)20-s + (0.893 − 0.448i)22-s + (−0.686 + 0.727i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2349 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.466 + 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2349 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.466 + 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2349\)    =    \(3^{4} \cdot 29\)
Sign: $0.466 + 0.884i$
Analytic conductor: \(10.9087\)
Root analytic conductor: \(10.9087\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2349} (1159, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2349,\ (0:\ ),\ 0.466 + 0.884i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2164275201 + 0.1306146776i\)
\(L(\frac12)\) \(\approx\) \(0.2164275201 + 0.1306146776i\)
\(L(1)\) \(\approx\) \(0.4614472403 + 0.02471201083i\)
\(L(1)\) \(\approx\) \(0.4614472403 + 0.02471201083i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.597 + 0.802i)T \)
5 \( 1 + (-0.835 - 0.549i)T \)
7 \( 1 + (-0.686 - 0.727i)T \)
11 \( 1 + (-0.893 - 0.448i)T \)
13 \( 1 + (0.396 - 0.918i)T \)
17 \( 1 + (-0.766 - 0.642i)T \)
19 \( 1 + (-0.766 + 0.642i)T \)
23 \( 1 + (-0.686 + 0.727i)T \)
31 \( 1 + (-0.973 - 0.230i)T \)
37 \( 1 + (-0.173 + 0.984i)T \)
41 \( 1 + (-0.597 - 0.802i)T \)
43 \( 1 + (0.0581 + 0.998i)T \)
47 \( 1 + (-0.973 + 0.230i)T \)
53 \( 1 + (-0.5 + 0.866i)T \)
59 \( 1 + (0.893 - 0.448i)T \)
61 \( 1 + (0.286 - 0.957i)T \)
67 \( 1 + (-0.993 + 0.116i)T \)
71 \( 1 + (-0.939 + 0.342i)T \)
73 \( 1 + (0.939 + 0.342i)T \)
79 \( 1 + (-0.597 + 0.802i)T \)
83 \( 1 + (0.597 - 0.802i)T \)
89 \( 1 + (0.939 + 0.342i)T \)
97 \( 1 + (0.835 - 0.549i)T \)
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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

−19.45556471911055830256423434016, −18.816010600642122842055679695455, −18.22302043681672476243478720410, −17.68261407430507463134341809261, −16.440781197478585910247428785656, −16.08535891000124558354556578324, −15.26790978066749010348996924938, −14.5450356680399316918243507819, −13.3037198471539204613262515400, −12.86066088657363831466218556716, −12.05893309424470361482593686248, −11.45858867045046068805304311105, −10.642032541136140165953864731040, −10.19284681149769809770516397059, −9.04344787713461216169479999321, −8.66047494588526405884033267536, −7.7946062872906804871757907063, −6.91126398908584561644260631902, −6.32616388971000508201014130415, −4.92070728833941087869239649517, −4.07648047295050347424389654030, −3.41679097714427012516888273915, −2.394817294315868615283920921918, −1.987812379070159477643347554837, −0.198124288239159234568378349830, 0.51286446159482102656689716255, 1.64150336260606837685341434384, 3.07645126871170184818613522761, 3.93084905562296765631716162052, 4.80049057219838221176643259322, 5.60766981280689509804634485507, 6.40421205110317335894717505467, 7.30561969665324179590039852462, 7.928345043185523406619623707171, 8.44535596693125013106897163172, 9.34043816878636489867807991711, 10.15769385599258175782789241902, 10.79724290146663107993819682245, 11.51196149447118930127482198002, 12.80515850311057067131696185988, 13.19489365166666680492593356547, 13.97339034577841440709302019627, 14.98906724004320979506144399255, 15.749231971734104177674736954606, 16.017964486908220381864795388533, 16.73473145627856597783720929421, 17.46056940655492112256598047476, 18.27659617546007983624145188101, 18.94791099337005004139701486688, 19.62305398852316016624155389702

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