Properties

Label 2-35-35.24-c4-0-2
Degree $2$
Conductor $35$
Sign $-0.846 - 0.532i$
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 + (−22.7 − 10.2i)5-s + 2.85i·6-s + (−47.2 − 12.8i)7-s + 20.9i·8-s + (31.2 + 54.1i)9-s + (−16.5 + 1.65i)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 + (−0.911 − 0.411i)5-s + 0.0793i·6-s + (−0.965 − 0.261i)7-s + 0.327i·8-s + (0.386 + 0.668i)9-s + (−0.165 + 0.0165i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.846 - 0.532i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.846 - 0.532i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $-0.846 - 0.532i$
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: \((2,\ 35,\ (\ :2),\ -0.846 - 0.532i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.163444 + 0.566148i\)
\(L(\frac12)\) \(\approx\) \(0.163444 + 0.566148i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (22.7 + 10.2i)T \)
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

−16.19216119278906030219258245607, −15.50694226880921643275093479802, −13.47907965828746619009783797906, −12.76632124121174136937096524178, −11.51610480170789576840147757839, −10.01976333980547177042991187820, −8.518725374761255307419945671081, −7.24295207890814963753340838717, −4.82336884218153083531390348421, −3.58103586288504999925732311676, 0.41428378553097452021146241514, 3.67429918623019110380666916287, 5.80931834861904551118545947022, 7.03090263830058683511237557434, 8.914867709185949013018518390236, 10.28635212503583973772326706123, 11.67008911902610568167416628582, 12.93356543298469226935928004106, 14.06503338790721954343078918328, 15.48494554564538337328399604303

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