↑

Properties

Label 1-304-304.91-r0-0-0
Degree $1$
Conductor $304$
Sign $0.634 + 0.772i$
Analytic cond. $1.41177$
Root an. cond. $1.41177$
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  + (0.342 + 0.939i)3-s + (0.984 + 0.173i)5-s + (−0.5 − 0.866i)7-s + (−0.766 + 0.642i)9-s + (0.866 + 0.5i)11-s + (0.342 − 0.939i)13-s + (0.173 + 0.984i)15-s + (0.766 + 0.642i)17-s + (0.642 − 0.766i)21-s + (0.173 + 0.984i)23-s + (0.939 + 0.342i)25-s + (−0.866 − 0.5i)27-s + (0.642 + 0.766i)29-s + (−0.5 − 0.866i)31-s + (−0.173 + 0.984i)33-s + ⋯
L(s)  = 1  + (0.342 + 0.939i)3-s + (0.984 + 0.173i)5-s + (−0.5 − 0.866i)7-s + (−0.766 + 0.642i)9-s + (0.866 + 0.5i)11-s + (0.342 − 0.939i)13-s + (0.173 + 0.984i)15-s + (0.766 + 0.642i)17-s + (0.642 − 0.766i)21-s + (0.173 + 0.984i)23-s + (0.939 + 0.342i)25-s + (−0.866 − 0.5i)27-s + (0.642 + 0.766i)29-s + (−0.5 − 0.866i)31-s + (−0.173 + 0.984i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.634 + 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.634 + 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $0.634 + 0.772i$
Analytic conductor: \(1.41177\)
Root analytic conductor: \(1.41177\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 304,\ (0:\ ),\ 0.634 + 0.772i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.477799643 + 0.6985714258i\)
\(L(\frac12)\) \(\approx\) \(1.477799643 + 0.6985714258i\)
\(L(1)\) \(\approx\) \(1.296683056 + 0.3772689666i\)
\(L(1)\) \(\approx\) \(1.296683056 + 0.3772689666i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 \)
good3 \( 1 + (0.342 + 0.939i)T \)
5 \( 1 + (0.984 + 0.173i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (0.866 + 0.5i)T \)
13 \( 1 + (0.342 - 0.939i)T \)
17 \( 1 + (0.766 + 0.642i)T \)
23 \( 1 + (0.173 + 0.984i)T \)
29 \( 1 + (0.642 + 0.766i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 - iT \)
41 \( 1 + (-0.939 + 0.342i)T \)
43 \( 1 + (-0.984 - 0.173i)T \)
47 \( 1 + (-0.766 + 0.642i)T \)
53 \( 1 + (0.984 - 0.173i)T \)
59 \( 1 + (-0.642 + 0.766i)T \)
61 \( 1 + (0.984 - 0.173i)T \)
67 \( 1 + (-0.642 - 0.766i)T \)
71 \( 1 + (-0.173 + 0.984i)T \)
73 \( 1 + (0.939 - 0.342i)T \)
79 \( 1 + (-0.939 + 0.342i)T \)
83 \( 1 + (0.866 - 0.5i)T \)
89 \( 1 + (-0.939 - 0.342i)T \)
97 \( 1 + (-0.766 - 0.642i)T \)
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

−25.07429656321364294405494084483, −24.63923281018839913146471822522, −23.54428376360158012473323577412, −22.47382351874627517014567418238, −21.59537064146627476717349658271, −20.74000652394890352147318992556, −19.62110034702588971878668021282, −18.7178614381327328509557089005, −18.25530462319040811778144894754, −17.03001379129183565560780090182, −16.32640696404834979491857866614, −14.81428028820690534375246751046, −14.01621216693891031650167948751, −13.328874337099475336297135337504, −12.243912712494615799843169680270, −11.62062580562591187039058518672, −9.969116351471226786617706495543, −9.01679478958998854500000335130, −8.448390963399170316712474714417, −6.735330242597475340370073057644, −6.326216815843193260221050488463, −5.17964669103489765239613263993, −3.3653522191101938147576646984, −2.31630042333595748951924426259, −1.24878054586017052812180858706, 1.51813224293297110539468948718, 3.09287860940263890232972576889, 3.88181592621671457536592227973, 5.207813558176154102844712816931, 6.18010696321616449214046221805, 7.41387621435641104481960672102, 8.71862878428937607714413241238, 9.78993223449950632633253565911, 10.18848958824144611369090550988, 11.194440166163415663478035677389, 12.73893066406407245525827173928, 13.6408710634729880249417668365, 14.4734119784996320427601508499, 15.26921226411216195452978054823, 16.526404464118857764922649873538, 17.0859105093338139519533172703, 17.999768746764601728948710200381, 19.4696892123640374119378253296, 20.12088889708903509803959491120, 20.98731477522197288723420484714, 21.84544734935000239452912359188, 22.605452573346128480615257684860, 23.40652473417913613826223417122, 24.96363205605306300306436899609, 25.60293464889225717427612516515

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