Properties

Label 2-35-7.6-c4-0-9
Degree $2$
Conductor $35$
Sign $-0.562 + 0.826i$
Analytic cond. $3.61794$
Root an. cond. $1.90209$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 1.17·2-s − 9.10i·3-s − 14.6·4-s − 11.1i·5-s − 10.6i·6-s + (−40.5 − 27.5i)7-s − 35.9·8-s − 1.87·9-s − 13.1i·10-s + 106.·11-s + 133. i·12-s − 243. i·13-s + (−47.5 − 32.3i)14-s − 101.·15-s + 191.·16-s + 148. i·17-s + ⋯
L(s)  = 1  + 0.293·2-s − 1.01i·3-s − 0.913·4-s − 0.447i·5-s − 0.296i·6-s + (−0.826 − 0.562i)7-s − 0.561·8-s − 0.0231·9-s − 0.131i·10-s + 0.882·11-s + 0.924i·12-s − 1.43i·13-s + (−0.242 − 0.165i)14-s − 0.452·15-s + 0.749·16-s + 0.512i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.562 + 0.826i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.562 + 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $-0.562 + 0.826i$
Analytic conductor: \(3.61794\)
Root analytic conductor: \(1.90209\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{35} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 35,\ (\ :2),\ -0.562 + 0.826i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.527671 - 0.997766i\)
\(L(\frac12)\) \(\approx\) \(0.527671 - 0.997766i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + 11.1iT \)
7 \( 1 + (40.5 + 27.5i)T \)
good2 \( 1 - 1.17T + 16T^{2} \)
3 \( 1 + 9.10iT - 81T^{2} \)
11 \( 1 - 106.T + 1.46e4T^{2} \)
13 \( 1 + 243. iT - 2.85e4T^{2} \)
17 \( 1 - 148. iT - 8.35e4T^{2} \)
19 \( 1 - 244. iT - 1.30e5T^{2} \)
23 \( 1 - 621.T + 2.79e5T^{2} \)
29 \( 1 + 774.T + 7.07e5T^{2} \)
31 \( 1 + 403. iT - 9.23e5T^{2} \)
37 \( 1 + 1.29e3T + 1.87e6T^{2} \)
41 \( 1 + 2.53e3iT - 2.82e6T^{2} \)
43 \( 1 - 2.24e3T + 3.41e6T^{2} \)
47 \( 1 + 2.49e3iT - 4.87e6T^{2} \)
53 \( 1 + 281.T + 7.89e6T^{2} \)
59 \( 1 - 5.82e3iT - 1.21e7T^{2} \)
61 \( 1 - 6.26e3iT - 1.38e7T^{2} \)
67 \( 1 - 4.12e3T + 2.01e7T^{2} \)
71 \( 1 + 793.T + 2.54e7T^{2} \)
73 \( 1 - 1.60e3iT - 2.83e7T^{2} \)
79 \( 1 - 9.85e3T + 3.89e7T^{2} \)
83 \( 1 - 617. iT - 4.74e7T^{2} \)
89 \( 1 + 7.34e3iT - 6.27e7T^{2} \)
97 \( 1 - 67.6iT - 8.85e7T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−15.17151033356563211842518504787, −13.77198477773281493640313458290, −12.94251272915953340970913531935, −12.35181653281157088395893898906, −10.21992354529943637597600609945, −8.835551765556975416027430990784, −7.37171025043010950617173424372, −5.78131771745191043063098567108, −3.79427199849119524054647088234, −0.798391254824136071476031379032, 3.51599073321578748746783719278, 4.83013286727816391897178659131, 6.64328634492241439247201969979, 9.209524762779134939462511394657, 9.472653649293042801833040397919, 11.22534652834666613787050653274, 12.66926849913048319293804949473, 14.00317722463341272851007355457, 14.95817595953568866956050631528, 16.04874658594710082500866030796

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