↑

Properties

Label 1-3328-3328.3203-r0-0-0
Degree $1$
Conductor $3328$
Sign $0.170 - 0.985i$
Analytic cond. $15.4551$
Root an. cond. $15.4551$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.634 + 0.773i)3-s + (0.290 − 0.956i)5-s + (0.195 + 0.980i)7-s + (−0.195 + 0.980i)9-s + (−0.0980 − 0.995i)11-s + (0.923 − 0.382i)15-s + (0.382 − 0.923i)17-s + (−0.471 + 0.881i)19-s + (−0.634 + 0.773i)21-s + (−0.555 − 0.831i)23-s + (−0.831 − 0.555i)25-s + (−0.881 + 0.471i)27-s + (0.0980 − 0.995i)29-s + (−0.707 − 0.707i)31-s + (0.707 − 0.707i)33-s + ⋯
L(s)  = 1  + (0.634 + 0.773i)3-s + (0.290 − 0.956i)5-s + (0.195 + 0.980i)7-s + (−0.195 + 0.980i)9-s + (−0.0980 − 0.995i)11-s + (0.923 − 0.382i)15-s + (0.382 − 0.923i)17-s + (−0.471 + 0.881i)19-s + (−0.634 + 0.773i)21-s + (−0.555 − 0.831i)23-s + (−0.831 − 0.555i)25-s + (−0.881 + 0.471i)27-s + (0.0980 − 0.995i)29-s + (−0.707 − 0.707i)31-s + (0.707 − 0.707i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.170 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.170 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3328\)    =    \(2^{8} \cdot 13\)
Sign: $0.170 - 0.985i$
Analytic conductor: \(15.4551\)
Root analytic conductor: \(15.4551\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3328} (3203, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3328,\ (0:\ ),\ 0.170 - 0.985i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.114786280 - 0.9385080595i\)
\(L(\frac12)\) \(\approx\) \(1.114786280 - 0.9385080595i\)
\(L(1)\) \(\approx\) \(1.200793703 + 0.005571565617i\)
\(L(1)\) \(\approx\) \(1.200793703 + 0.005571565617i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 \)
good3 \( 1 + (0.634 + 0.773i)T \)
5 \( 1 + (0.290 - 0.956i)T \)
7 \( 1 + (0.195 + 0.980i)T \)
11 \( 1 + (-0.0980 - 0.995i)T \)
17 \( 1 + (0.382 - 0.923i)T \)
19 \( 1 + (-0.471 + 0.881i)T \)
23 \( 1 + (-0.555 - 0.831i)T \)
29 \( 1 + (0.0980 - 0.995i)T \)
31 \( 1 + (-0.707 - 0.707i)T \)
37 \( 1 + (-0.881 + 0.471i)T \)
41 \( 1 + (-0.555 - 0.831i)T \)
43 \( 1 + (0.634 - 0.773i)T \)
47 \( 1 + (0.382 - 0.923i)T \)
53 \( 1 + (0.0980 + 0.995i)T \)
59 \( 1 + (-0.956 - 0.290i)T \)
61 \( 1 + (0.773 - 0.634i)T \)
67 \( 1 + (-0.634 - 0.773i)T \)
71 \( 1 + (-0.980 + 0.195i)T \)
73 \( 1 + (0.195 - 0.980i)T \)
79 \( 1 + (0.923 - 0.382i)T \)
83 \( 1 + (0.881 + 0.471i)T \)
89 \( 1 + (-0.831 - 0.555i)T \)
97 \( 1 + (-0.707 - 0.707i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.22480896313437205490669286387, −18.13687818741762282162178241375, −17.70984908772588498991106921047, −17.35097998160975006141302835293, −16.22619132934127512622757482531, −15.23693950557771536654257807114, −14.68364688967374624695583995436, −14.22501479932572222423152657109, −13.468349726461307383532131561587, −12.88988853994100547018043808060, −12.17010770100544784241206392494, −11.196087745853793894144039987169, −10.55560132708719000054714645874, −9.88219848241345410938868422564, −9.12359114191850662698152840894, −8.15144560165416920825665045402, −7.44373243231925222041049223991, −6.99823931636735256079169435471, −6.39683863590863949333712229446, −5.40016791639918947620721361790, −4.257689098480086211520950420504, −3.55086091097143001882184507686, −2.79568271873031814277978008863, −1.84686364732065576900625075754, −1.32939111022055179590185652526, 0.354506300445220906878036253189, 1.80541069069396211453481019969, 2.386400499437595795177138846937, 3.34802453287164068909917035040, 4.1361306175138454621779138393, 4.97617014121653436463615782711, 5.59507835255272497152543532792, 6.15629277323913447337146329521, 7.63187099444937566732894286507, 8.28966129549989675981947057986, 8.80104950735100726439947730250, 9.34354499975991315207930856181, 10.124155593840037247341427569403, 10.840778976755071642699774172550, 11.87714806046523255973189485969, 12.26947735443431442084586486339, 13.317643457038986798279588952542, 13.89433993607985497769760012452, 14.46772578320616631638562011545, 15.46291306465816299823674790, 15.78814491424221327446439187799, 16.739755882437964103953363510646, 16.87074173155368383211640596923, 18.1613778649235660801040066978, 18.80971774988873516652585216495

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