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Properties

Label 2-35-35.24-c4-0-5
Degree $2$
Conductor $35$
Sign $0.129 - 0.991i$
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  + (4.03 − 2.32i)2-s + (−7.86 + 13.6i)3-s + (2.83 − 4.90i)4-s + (−15.2 + 19.8i)5-s + 73.2i·6-s + (45.1 − 19.0i)7-s + 48.1i·8-s + (−83.2 − 144. i)9-s + (−15.2 + 115. i)10-s + (21.2 − 36.7i)11-s + (44.5 + 77.1i)12-s + 247.·13-s + (137. − 181. i)14-s + (−150. − 363. i)15-s + (157. + 272. i)16-s + (−152. + 263. i)17-s + ⋯
L(s)  = 1  + (1.00 − 0.581i)2-s + (−0.874 + 1.51i)3-s + (0.177 − 0.306i)4-s + (−0.609 + 0.792i)5-s + 2.03i·6-s + (0.921 − 0.388i)7-s + 0.751i·8-s + (−1.02 − 1.78i)9-s + (−0.152 + 1.15i)10-s + (0.175 − 0.303i)11-s + (0.309 + 0.536i)12-s + 1.46·13-s + (0.702 − 0.927i)14-s + (−0.668 − 1.61i)15-s + (0.614 + 1.06i)16-s + (−0.526 + 0.912i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.129 - 0.991i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.129 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $0.129 - 0.991i$
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: \((2,\ 35,\ (\ :2),\ 0.129 - 0.991i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.24366 + 1.09179i\)
\(L(\frac12)\) \(\approx\) \(1.24366 + 1.09179i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (15.2 - 19.8i)T \)
7 \( 1 + (-45.1 + 19.0i)T \)
good2 \( 1 + (-4.03 + 2.32i)T + (8 - 13.8i)T^{2} \)
3 \( 1 + (7.86 - 13.6i)T + (-40.5 - 70.1i)T^{2} \)
11 \( 1 + (-21.2 + 36.7i)T + (-7.32e3 - 1.26e4i)T^{2} \)
13 \( 1 - 247.T + 2.85e4T^{2} \)
17 \( 1 + (152. - 263. i)T + (-4.17e4 - 7.23e4i)T^{2} \)
19 \( 1 + (10.5 - 6.09i)T + (6.51e4 - 1.12e5i)T^{2} \)
23 \( 1 + (32.1 - 18.5i)T + (1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 - 1.09e3T + 7.07e5T^{2} \)
31 \( 1 + (502. + 290. i)T + (4.61e5 + 7.99e5i)T^{2} \)
37 \( 1 + (-362. + 209. i)T + (9.37e5 - 1.62e6i)T^{2} \)
41 \( 1 + 1.14e3iT - 2.82e6T^{2} \)
43 \( 1 + 354. iT - 3.41e6T^{2} \)
47 \( 1 + (-426. - 738. i)T + (-2.43e6 + 4.22e6i)T^{2} \)
53 \( 1 + (-2.94e3 - 1.70e3i)T + (3.94e6 + 6.83e6i)T^{2} \)
59 \( 1 + (4.39e3 + 2.53e3i)T + (6.05e6 + 1.04e7i)T^{2} \)
61 \( 1 + (-165. + 95.5i)T + (6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (1.48e3 + 857. i)T + (1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + 1.02e3T + 2.54e7T^{2} \)
73 \( 1 + (-3.35e3 + 5.81e3i)T + (-1.41e7 - 2.45e7i)T^{2} \)
79 \( 1 + (-276. - 479. i)T + (-1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 + 5.18e3T + 4.74e7T^{2} \)
89 \( 1 + (8.12e3 - 4.69e3i)T + (3.13e7 - 5.43e7i)T^{2} \)
97 \( 1 + 333.T + 8.85e7T^{2} \)
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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

−15.76093289512625379086481869415, −14.88109449233189736355493469191, −13.83663973858970624339275244408, −11.99041233782737048186674242054, −10.96665365341393837001162663010, −10.73170304999458767950610677311, −8.483196080849459443205657544694, −5.98616509636125836057181426756, −4.43532362717042888904603299382, −3.63533724752094230902637303743, 1.10087638631693547303069277853, 4.70155169373622505454136687502, 5.88167269550442926096222887979, 7.15640386591954416700170154024, 8.489590464682450384948095032987, 11.30252028243878047496355418765, 12.14887621128019144893084308428, 13.12274216328582238381121016379, 13.98173173755064913261929088559, 15.51007220159599650135675253822

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