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Properties

Label 2-35-35.24-c4-0-1
Degree $2$
Conductor $35$
Sign $-0.513 + 0.858i$
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  + (−4.29 + 2.48i)2-s + (−5.01 + 8.67i)3-s + (4.31 − 7.46i)4-s + (4.41 + 24.6i)5-s − 49.7i·6-s + (−32.5 − 36.5i)7-s − 36.5i·8-s + (−9.71 − 16.8i)9-s + (−80.0 − 94.8i)10-s + (39.0 − 67.6i)11-s + (43.2 + 74.8i)12-s − 90.2·13-s + (230. + 76.4i)14-s + (−235. − 85.0i)15-s + (159. + 276. i)16-s + (−105. + 181. i)17-s + ⋯
L(s)  = 1  + (−1.07 + 0.620i)2-s + (−0.556 + 0.964i)3-s + (0.269 − 0.466i)4-s + (0.176 + 0.984i)5-s − 1.38i·6-s + (−0.664 − 0.746i)7-s − 0.571i·8-s + (−0.119 − 0.207i)9-s + (−0.800 − 0.948i)10-s + (0.322 − 0.559i)11-s + (0.300 + 0.519i)12-s − 0.534·13-s + (1.17 + 0.389i)14-s + (−1.04 − 0.377i)15-s + (0.624 + 1.08i)16-s + (−0.363 + 0.629i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.513 + 0.858i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.513 + 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $-0.513 + 0.858i$
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: \((2,\ 35,\ (\ :2),\ -0.513 + 0.858i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.121778 - 0.214793i\)
\(L(\frac12)\) \(\approx\) \(0.121778 - 0.214793i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-4.41 - 24.6i)T \)
7 \( 1 + (32.5 + 36.5i)T \)
good2 \( 1 + (4.29 - 2.48i)T + (8 - 13.8i)T^{2} \)
3 \( 1 + (5.01 - 8.67i)T + (-40.5 - 70.1i)T^{2} \)
11 \( 1 + (-39.0 + 67.6i)T + (-7.32e3 - 1.26e4i)T^{2} \)
13 \( 1 + 90.2T + 2.85e4T^{2} \)
17 \( 1 + (105. - 181. i)T + (-4.17e4 - 7.23e4i)T^{2} \)
19 \( 1 + (325. - 188. i)T + (6.51e4 - 1.12e5i)T^{2} \)
23 \( 1 + (-679. + 392. i)T + (1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + 1.38e3T + 7.07e5T^{2} \)
31 \( 1 + (204. + 117. i)T + (4.61e5 + 7.99e5i)T^{2} \)
37 \( 1 + (961. - 555. i)T + (9.37e5 - 1.62e6i)T^{2} \)
41 \( 1 + 1.24e3iT - 2.82e6T^{2} \)
43 \( 1 - 469. iT - 3.41e6T^{2} \)
47 \( 1 + (-2.02e3 - 3.50e3i)T + (-2.43e6 + 4.22e6i)T^{2} \)
53 \( 1 + (526. + 303. i)T + (3.94e6 + 6.83e6i)T^{2} \)
59 \( 1 + (-3.55e3 - 2.05e3i)T + (6.05e6 + 1.04e7i)T^{2} \)
61 \( 1 + (2.50e3 - 1.44e3i)T + (6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (3.05e3 + 1.76e3i)T + (1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 - 5.95e3T + 2.54e7T^{2} \)
73 \( 1 + (1.48e3 - 2.57e3i)T + (-1.41e7 - 2.45e7i)T^{2} \)
79 \( 1 + (2.46e3 + 4.26e3i)T + (-1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 + 3.36e3T + 4.74e7T^{2} \)
89 \( 1 + (5.32e3 - 3.07e3i)T + (3.13e7 - 5.43e7i)T^{2} \)
97 \( 1 - 1.69e4T + 8.85e7T^{2} \)
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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

−16.94038311452860671251975019675, −15.72917379076018232809455502881, −14.70835256773937162514268222993, −13.03589015681938048011044277172, −10.88728727636294203821915198426, −10.30882622402002839937167059488, −9.165918884249085037405015756635, −7.38293545030738685253738786250, −6.21175096055146415924637373749, −3.86018401204051870167504971831, 0.25598980508054681868776587611, 1.91915388958935912991190270084, 5.40099212013896414231733412511, 7.13226805482360420422992602235, 8.880478697501054034457688362342, 9.608268964479529512646672741614, 11.40049462894527572604373260822, 12.35773928812644021414138799377, 13.19656666357153806141462413945, 15.19599531159040190621984052272

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