↑

Properties

Label 2-35-35.34-c4-0-1
Degree $2$
Conductor $35$
Sign $-0.357 + 0.933i$
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  + 7.21i·2-s − 9.48·3-s − 36.0·4-s + (22.2 − 11.4i)5-s − 68.4i·6-s + (−36.4 + 32.7i)7-s − 144. i·8-s + 9·9-s + (82.3 + 160. i)10-s + 40.4·11-s + 342.·12-s − 222.·13-s + (−236. − 263. i)14-s + (−211. + 108. i)15-s + 467.·16-s − 227.·17-s + ⋯
L(s)  = 1  + 1.80i·2-s − 1.05·3-s − 2.25·4-s + (0.889 − 0.456i)5-s − 1.90i·6-s + (−0.744 + 0.667i)7-s − 2.26i·8-s + 0.111·9-s + (0.823 + 1.60i)10-s + 0.333·11-s + 2.37·12-s − 1.31·13-s + (−1.20 − 1.34i)14-s + (−0.937 + 0.481i)15-s + 1.82·16-s − 0.786·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.357 + 0.933i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.357 + 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $-0.357 + 0.933i$
Analytic conductor: \(3.61794\)
Root analytic conductor: \(1.90209\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{35} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 35,\ (\ :2),\ -0.357 + 0.933i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.235323 - 0.342099i\)
\(L(\frac12)\) \(\approx\) \(0.235323 - 0.342099i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-22.2 + 11.4i)T \)
7 \( 1 + (36.4 - 32.7i)T \)
good2 \( 1 - 7.21iT - 16T^{2} \)
3 \( 1 + 9.48T + 81T^{2} \)
11 \( 1 - 40.4T + 1.46e4T^{2} \)
13 \( 1 + 222.T + 2.85e4T^{2} \)
17 \( 1 + 227.T + 8.35e4T^{2} \)
19 \( 1 - 180. iT - 1.30e5T^{2} \)
23 \( 1 - 1.00e3iT - 2.79e5T^{2} \)
29 \( 1 - 50.2T + 7.07e5T^{2} \)
31 \( 1 - 1.28e3iT - 9.23e5T^{2} \)
37 \( 1 + 312. iT - 1.87e6T^{2} \)
41 \( 1 - 1.02e3iT - 2.82e6T^{2} \)
43 \( 1 + 2.04e3iT - 3.41e6T^{2} \)
47 \( 1 + 794.T + 4.87e6T^{2} \)
53 \( 1 + 1.04e3iT - 7.89e6T^{2} \)
59 \( 1 + 1.29e3iT - 1.21e7T^{2} \)
61 \( 1 - 4.52e3iT - 1.38e7T^{2} \)
67 \( 1 - 4.83e3iT - 2.01e7T^{2} \)
71 \( 1 + 4.93e3T + 2.54e7T^{2} \)
73 \( 1 - 1.95e3T + 2.83e7T^{2} \)
79 \( 1 - 1.89e3T + 3.89e7T^{2} \)
83 \( 1 - 77.7T + 4.74e7T^{2} \)
89 \( 1 - 320. iT - 6.27e7T^{2} \)
97 \( 1 - 1.26e4T + 8.85e7T^{2} \)
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

−16.63532786656629156034300358242, −15.75291572923556220262944491298, −14.49865447495834550044787928459, −13.27436435624196374470711520564, −12.09432233769644712049557443253, −9.853596010456915092882665157236, −8.835521083176348160329558652826, −6.94580291500696261808412415664, −5.86901721906674966994327014228, −5.08021399825307536930319945838, 0.32141121756519341972011670848, 2.59016448918214736749619526038, 4.66170814000672037618657579480, 6.49734034705987263069442121363, 9.340402192314711851909617316588, 10.30720971521995658594956879139, 11.09141444351906080009389146035, 12.30834297509211741313814508525, 13.24237558892929553194060586473, 14.42309292817104676999772421566

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