↑

Properties

Label 1-319-319.10-r0-0-0
Degree $1$
Conductor $319$
Sign $-0.853 + 0.521i$
Analytic cond. $1.48142$
Root an. cond. $1.48142$
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  + (−0.433 + 0.900i)2-s + (0.781 + 0.623i)3-s + (−0.623 − 0.781i)4-s + (0.900 + 0.433i)5-s + (−0.900 + 0.433i)6-s + (−0.623 + 0.781i)7-s + (0.974 − 0.222i)8-s + (0.222 + 0.974i)9-s + (−0.781 + 0.623i)10-s − i·12-s + (−0.222 + 0.974i)13-s + (−0.433 − 0.900i)14-s + (0.433 + 0.900i)15-s + (−0.222 + 0.974i)16-s − i·17-s + (−0.974 − 0.222i)18-s + ⋯
L(s)  = 1  + (−0.433 + 0.900i)2-s + (0.781 + 0.623i)3-s + (−0.623 − 0.781i)4-s + (0.900 + 0.433i)5-s + (−0.900 + 0.433i)6-s + (−0.623 + 0.781i)7-s + (0.974 − 0.222i)8-s + (0.222 + 0.974i)9-s + (−0.781 + 0.623i)10-s − i·12-s + (−0.222 + 0.974i)13-s + (−0.433 − 0.900i)14-s + (0.433 + 0.900i)15-s + (−0.222 + 0.974i)16-s − i·17-s + (−0.974 − 0.222i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 319 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.853 + 0.521i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 319 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.853 + 0.521i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(319\)    =    \(11 \cdot 29\)
Sign: $-0.853 + 0.521i$
Analytic conductor: \(1.48142\)
Root analytic conductor: \(1.48142\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{319} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 319,\ (0:\ ),\ -0.853 + 0.521i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3529600715 + 1.254131298i\)
\(L(\frac12)\) \(\approx\) \(0.3529600715 + 1.254131298i\)
\(L(1)\) \(\approx\) \(0.7692763753 + 0.8086545298i\)
\(L(1)\) \(\approx\) \(0.7692763753 + 0.8086545298i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.433 + 0.900i)T \)
3 \( 1 + (0.781 + 0.623i)T \)
5 \( 1 + (0.900 + 0.433i)T \)
7 \( 1 + (-0.623 + 0.781i)T \)
13 \( 1 + (-0.222 + 0.974i)T \)
17 \( 1 - iT \)
19 \( 1 + (0.781 - 0.623i)T \)
23 \( 1 + (-0.900 + 0.433i)T \)
31 \( 1 + (0.433 - 0.900i)T \)
37 \( 1 + (-0.974 + 0.222i)T \)
41 \( 1 + iT \)
43 \( 1 + (0.433 + 0.900i)T \)
47 \( 1 + (0.974 + 0.222i)T \)
53 \( 1 + (-0.900 - 0.433i)T \)
59 \( 1 + T \)
61 \( 1 + (-0.781 - 0.623i)T \)
67 \( 1 + (0.222 + 0.974i)T \)
71 \( 1 + (0.222 - 0.974i)T \)
73 \( 1 + (-0.433 - 0.900i)T \)
79 \( 1 + (-0.974 + 0.222i)T \)
83 \( 1 + (-0.623 - 0.781i)T \)
89 \( 1 + (0.433 - 0.900i)T \)
97 \( 1 + (0.781 - 0.623i)T \)
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

−25.04207476684340973871779897969, −24.01475395827071206832266561569, −22.82716109078162802254014657507, −21.935007089094350137295538214361, −20.802054035490924427200092847637, −20.28859216940405294054051864954, −19.57473379965921716482989792735, −18.63939576236626102479692013628, −17.66253180400903467270133205491, −17.11633945705958064351879292365, −15.853536714229150745929479111868, −14.176461701551902832922823798289, −13.7041491049133549157893420402, −12.67032047500674900627341518056, −12.31894688301142967820556441787, −10.43737490605532418337187459835, −10.0196390172163114971438432934, −8.93569821182328143486757365056, −8.09559913128448535524383034706, −7.05571757910898308806194193390, −5.69612155603500112971830026459, −4.04785699933730836650096602429, −3.09845580277616490084140226417, −1.963857656664691627547443525903, −0.91685113363123022106528840300, 1.93348262372945554630436639440, 3.03604170773066841505000125416, 4.59573445601925365626367851842, 5.60324446675486875793711147085, 6.62644778824183304287443333954, 7.64368358772397755372944612526, 8.97969502531370941613491740565, 9.48205996684280012863518646603, 10.080846092734776999482845598447, 11.48964365474974847574228561747, 13.24031110212641889287846173733, 13.937241809949079503555772054564, 14.63653203053287509958848681358, 15.693690496471133315850321249958, 16.207855951133635529142877817418, 17.33931663717584540182404587076, 18.40424961717881004821348588720, 19.00673623457785535182325402303, 19.99795976695568058933379720654, 21.21354485503807062268754140469, 22.122155339628694866958936078334, 22.58549195246271335400451128270, 24.17253949403403499344882479637, 24.89854729645083837768877228334, 25.64892570571564758004025143629

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