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Properties

Label 2-35-35.24-c4-0-4
Degree $2$
Conductor $35$
Sign $0.235 - 0.971i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.575 + 0.332i)2-s + (2.14 − 3.72i)3-s + (−7.77 + 13.4i)4-s + (−2.48 + 24.8i)5-s + 2.85i·6-s + (47.2 + 12.8i)7-s − 20.9i·8-s + (31.2 + 54.1i)9-s + (−6.84 − 15.1i)10-s + (−47.5 + 82.4i)11-s + (33.4 + 57.8i)12-s − 113.·13-s + (−31.5 + 8.34i)14-s + (87.2 + 62.6i)15-s + (−117. − 203. i)16-s + (105. − 183. i)17-s + ⋯
L(s)  = 1  + (−0.143 + 0.0831i)2-s + (0.238 − 0.413i)3-s + (−0.486 + 0.842i)4-s + (−0.0992 + 0.995i)5-s + 0.0793i·6-s + (0.965 + 0.261i)7-s − 0.327i·8-s + (0.386 + 0.668i)9-s + (−0.0684 − 0.151i)10-s + (−0.393 + 0.681i)11-s + (0.232 + 0.402i)12-s − 0.671·13-s + (−0.160 + 0.0425i)14-s + (0.387 + 0.278i)15-s + (−0.458 − 0.794i)16-s + (0.366 − 0.634i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.235 - 0.971i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.235 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $0.235 - 0.971i$
Analytic conductor: \(3.61794\)
Root analytic conductor: \(1.90209\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{35} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 35,\ (\ :2),\ 0.235 - 0.971i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.01893 + 0.801769i\)
\(L(\frac12)\) \(\approx\) \(1.01893 + 0.801769i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.48 - 24.8i)T \)
7 \( 1 + (-47.2 - 12.8i)T \)
good2 \( 1 + (0.575 - 0.332i)T + (8 - 13.8i)T^{2} \)
3 \( 1 + (-2.14 + 3.72i)T + (-40.5 - 70.1i)T^{2} \)
11 \( 1 + (47.5 - 82.4i)T + (-7.32e3 - 1.26e4i)T^{2} \)
13 \( 1 + 113.T + 2.85e4T^{2} \)
17 \( 1 + (-105. + 183. i)T + (-4.17e4 - 7.23e4i)T^{2} \)
19 \( 1 + (-287. + 165. i)T + (6.51e4 - 1.12e5i)T^{2} \)
23 \( 1 + (-103. + 60.0i)T + (1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + 450.T + 7.07e5T^{2} \)
31 \( 1 + (-511. - 295. i)T + (4.61e5 + 7.99e5i)T^{2} \)
37 \( 1 + (-2.32e3 + 1.34e3i)T + (9.37e5 - 1.62e6i)T^{2} \)
41 \( 1 - 1.52e3iT - 2.82e6T^{2} \)
43 \( 1 + 1.37e3iT - 3.41e6T^{2} \)
47 \( 1 + (-1.74e3 - 3.01e3i)T + (-2.43e6 + 4.22e6i)T^{2} \)
53 \( 1 + (3.30e3 + 1.90e3i)T + (3.94e6 + 6.83e6i)T^{2} \)
59 \( 1 + (2.16e3 + 1.24e3i)T + (6.05e6 + 1.04e7i)T^{2} \)
61 \( 1 + (-5.76e3 + 3.33e3i)T + (6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (2.85e3 + 1.64e3i)T + (1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + 6.05e3T + 2.54e7T^{2} \)
73 \( 1 + (-688. + 1.19e3i)T + (-1.41e7 - 2.45e7i)T^{2} \)
79 \( 1 + (9.97 + 17.2i)T + (-1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 - 7.24e3T + 4.74e7T^{2} \)
89 \( 1 + (818. - 472. i)T + (3.13e7 - 5.43e7i)T^{2} \)
97 \( 1 + 8.58e3T + 8.85e7T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−16.04339248150704985201522358993, −14.68736572018727521925467942972, −13.71318172160807198968840594734, −12.45149685318305524956724272703, −11.20862618203487193302915751775, −9.634621407348669712227168440187, −7.85423656973639389179977715728, −7.32221588136399260498316511450, −4.76376820334980041030765145944, −2.59856945359102693688109613450, 1.07381861664661345210578299793, 4.31802626275368274075173259425, 5.56223928904527726241271036283, 8.024722927058228756162188759625, 9.221015132066841054554345883432, 10.27937988517220306761368322232, 11.77258218682801581790955813383, 13.27161458015012392876148436454, 14.46776836345346559454099891091, 15.36825393929814980743494357344

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