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Properties

Label 2-35-35.24-c4-0-13
Degree $2$
Conductor $35$
Sign $0.152 + 0.988i$
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 + (−19.1 − 16.1i)5-s − 49.7i·6-s + (32.5 + 36.5i)7-s + 36.5i·8-s + (−9.71 − 16.8i)9-s + (−122. − 21.8i)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 + (−0.764 − 0.644i)5-s − 1.38i·6-s + (0.664 + 0.746i)7-s + 0.571i·8-s + (−0.119 − 0.207i)9-s + (−1.22 − 0.218i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.152 + 0.988i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.152 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $0.152 + 0.988i$
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: \((2,\ 35,\ (\ :2),\ 0.152 + 0.988i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.93963 - 1.66300i\)
\(L(\frac12)\) \(\approx\) \(1.93963 - 1.66300i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (19.1 + 16.1i)T \)
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

−15.04663979420604220699101268910, −13.98203441447162614748202144218, −13.02722176156275383491842027542, −12.10363823738277123766804647188, −11.29728847471430510573187683743, −8.682829586496963716060163435640, −7.82238763514010390126981515101, −5.54099659381881675756840601723, −3.82968317785477727032318832426, −1.90983660137416931043306899634, 3.75427902488487366826771339675, 4.44976495516692168184851805329, 6.55311884790193779308078029367, 8.015110308624538959823273120347, 9.875118147880944225771831448834, 11.08574896459442837433757040722, 12.74879309704253385614411452063, 14.28884308172791083089917255986, 14.75828273177260265840430627049, 15.55204772756027521283011049092

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