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Properties

Label 1-3168-3168.2797-r0-0-0
Degree $1$
Conductor $3168$
Sign $0.943 + 0.331i$
Analytic cond. $14.7121$
Root an. cond. $14.7121$
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  + (−0.838 + 0.544i)5-s + (0.994 + 0.104i)7-s + (−0.0523 + 0.998i)13-s + (−0.309 − 0.951i)17-s + (−0.987 − 0.156i)19-s + (−0.866 − 0.5i)23-s + (0.406 − 0.913i)25-s + (0.777 − 0.629i)29-s + (0.669 + 0.743i)31-s + (−0.891 + 0.453i)35-s + (−0.987 + 0.156i)37-s + (0.994 − 0.104i)41-s + (0.965 + 0.258i)43-s + (−0.913 − 0.406i)47-s + (0.978 + 0.207i)49-s + ⋯
L(s)  = 1  + (−0.838 + 0.544i)5-s + (0.994 + 0.104i)7-s + (−0.0523 + 0.998i)13-s + (−0.309 − 0.951i)17-s + (−0.987 − 0.156i)19-s + (−0.866 − 0.5i)23-s + (0.406 − 0.913i)25-s + (0.777 − 0.629i)29-s + (0.669 + 0.743i)31-s + (−0.891 + 0.453i)35-s + (−0.987 + 0.156i)37-s + (0.994 − 0.104i)41-s + (0.965 + 0.258i)43-s + (−0.913 − 0.406i)47-s + (0.978 + 0.207i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3168 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.943 + 0.331i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3168 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.943 + 0.331i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3168\)    =    \(2^{5} \cdot 3^{2} \cdot 11\)
Sign: $0.943 + 0.331i$
Analytic conductor: \(14.7121\)
Root analytic conductor: \(14.7121\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3168} (2797, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3168,\ (0:\ ),\ 0.943 + 0.331i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.421308177 + 0.2427936153i\)
\(L(\frac12)\) \(\approx\) \(1.421308177 + 0.2427936153i\)
\(L(1)\) \(\approx\) \(0.9961350440 + 0.1016950781i\)
\(L(1)\) \(\approx\) \(0.9961350440 + 0.1016950781i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
11 \( 1 \)
good5 \( 1 + (-0.838 + 0.544i)T \)
7 \( 1 + (0.994 + 0.104i)T \)
13 \( 1 + (-0.0523 + 0.998i)T \)
17 \( 1 + (-0.309 - 0.951i)T \)
19 \( 1 + (-0.987 - 0.156i)T \)
23 \( 1 + (-0.866 - 0.5i)T \)
29 \( 1 + (0.777 - 0.629i)T \)
31 \( 1 + (0.669 + 0.743i)T \)
37 \( 1 + (-0.987 + 0.156i)T \)
41 \( 1 + (0.994 - 0.104i)T \)
43 \( 1 + (0.965 + 0.258i)T \)
47 \( 1 + (-0.913 - 0.406i)T \)
53 \( 1 + (0.453 + 0.891i)T \)
59 \( 1 + (0.358 - 0.933i)T \)
61 \( 1 + (-0.998 + 0.0523i)T \)
67 \( 1 + (0.965 - 0.258i)T \)
71 \( 1 + (0.951 - 0.309i)T \)
73 \( 1 + (0.587 - 0.809i)T \)
79 \( 1 + (0.978 - 0.207i)T \)
83 \( 1 + (0.998 - 0.0523i)T \)
89 \( 1 - iT \)
97 \( 1 + (-0.978 + 0.207i)T \)
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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

−19.135623710617195628902632101931, −18.022917866883487295791869641653, −17.55799889177015945347104948713, −16.9284913190791707388628311875, −16.05672593019841284916665352598, −15.36923864868375071068318658743, −14.90191133421387374576750947090, −14.10899744024548595680183511152, −13.21930241088254402325922384860, −12.4755842968669880420539731978, −12.02811623461737563519783077772, −11.013432938997482317543627985723, −10.7022728304838503034370971367, −9.712982428506117758744189172979, −8.63014851747014398579959931493, −8.18922343252225566390185500836, −7.737041403679822137374698423090, −6.73068568947043287823813701347, −5.75299441255108529067984512576, −5.049826463852322162569690404296, −4.21004355687082221303505018099, −3.74564949355156156325403849375, −2.53576180524158813873885484496, −1.61318216660227220105416427942, −0.684417386699754838877900856617, 0.68120204160929589997695561720, 1.989009110043129839682644976235, 2.55924801181102070229715910997, 3.67247157998575940510931857566, 4.50612045973740189048715433881, 4.8406701509339504881258666854, 6.19146021667518181087620252608, 6.76842711665441325480572011242, 7.56899220708474745334389018025, 8.26216943029299046106684990101, 8.84740383093716186676101609252, 9.8162723769033831782785851300, 10.7998786157853836705746430243, 11.13883484379040343731584224963, 12.07360720497154390795299772708, 12.26741922051803162440979785556, 13.72418307240809260402035338178, 14.109360815327057823196807999111, 14.77981615410577798477324278576, 15.55128077175379869472803171283, 16.061590893728248664477991283674, 16.92170173292713968140217547100, 17.76520864652520136739763768281, 18.249915938656463144939191621650, 19.06982771030759503120676448843

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