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Properties

Label 2-35-35.19-c4-0-2
Degree $2$
Conductor $35$
Sign $0.664 - 0.747i$
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  + (−2.10 − 1.21i)2-s + (3.73 + 6.46i)3-s + (−5.05 − 8.75i)4-s + (11.6 + 22.1i)5-s − 18.1i·6-s + (22.9 + 43.3i)7-s + 63.3i·8-s + (12.6 − 21.8i)9-s + (2.35 − 60.6i)10-s + (95.0 + 164. i)11-s + (37.7 − 65.3i)12-s + 63.2·13-s + (4.42 − 118. i)14-s + (−99.5 + 157. i)15-s + (−3.93 + 6.81i)16-s + (−166. − 288. i)17-s + ⋯
L(s)  = 1  + (−0.525 − 0.303i)2-s + (0.414 + 0.718i)3-s + (−0.315 − 0.547i)4-s + (0.466 + 0.884i)5-s − 0.503i·6-s + (0.467 + 0.884i)7-s + 0.990i·8-s + (0.155 − 0.269i)9-s + (0.0235 − 0.606i)10-s + (0.785 + 1.36i)11-s + (0.262 − 0.453i)12-s + 0.374·13-s + (0.0225 − 0.606i)14-s + (−0.442 + 0.701i)15-s + (−0.0153 + 0.0266i)16-s + (−0.575 − 0.996i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.664 - 0.747i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.664 - 0.747i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $0.664 - 0.747i$
Analytic conductor: \(3.61794\)
Root analytic conductor: \(1.90209\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{35} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 35,\ (\ :2),\ 0.664 - 0.747i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.15641 + 0.519527i\)
\(L(\frac12)\) \(\approx\) \(1.15641 + 0.519527i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-11.6 - 22.1i)T \)
7 \( 1 + (-22.9 - 43.3i)T \)
good2 \( 1 + (2.10 + 1.21i)T + (8 + 13.8i)T^{2} \)
3 \( 1 + (-3.73 - 6.46i)T + (-40.5 + 70.1i)T^{2} \)
11 \( 1 + (-95.0 - 164. i)T + (-7.32e3 + 1.26e4i)T^{2} \)
13 \( 1 - 63.2T + 2.85e4T^{2} \)
17 \( 1 + (166. + 288. i)T + (-4.17e4 + 7.23e4i)T^{2} \)
19 \( 1 + (343. + 198. i)T + (6.51e4 + 1.12e5i)T^{2} \)
23 \( 1 + (361. + 208. i)T + (1.39e5 + 2.42e5i)T^{2} \)
29 \( 1 - 1.38e3T + 7.07e5T^{2} \)
31 \( 1 + (-416. + 240. i)T + (4.61e5 - 7.99e5i)T^{2} \)
37 \( 1 + (1.36e3 + 788. i)T + (9.37e5 + 1.62e6i)T^{2} \)
41 \( 1 + 959. iT - 2.82e6T^{2} \)
43 \( 1 - 1.68e3iT - 3.41e6T^{2} \)
47 \( 1 + (-868. + 1.50e3i)T + (-2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + (-2.06e3 + 1.19e3i)T + (3.94e6 - 6.83e6i)T^{2} \)
59 \( 1 + (-131. + 76.0i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (2.35e3 + 1.35e3i)T + (6.92e6 + 1.19e7i)T^{2} \)
67 \( 1 + (-2.68e3 + 1.55e3i)T + (1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 + 739.T + 2.54e7T^{2} \)
73 \( 1 + (2.97e3 + 5.14e3i)T + (-1.41e7 + 2.45e7i)T^{2} \)
79 \( 1 + (-1.71e3 + 2.97e3i)T + (-1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 - 1.09e4T + 4.74e7T^{2} \)
89 \( 1 + (2.30e3 + 1.32e3i)T + (3.13e7 + 5.43e7i)T^{2} \)
97 \( 1 + 5.79e3T + 8.85e7T^{2} \)
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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

−15.51823856908033037109892492188, −14.85092536823310166402760803188, −13.99590439339910975879586079220, −11.97975044876825466935341961098, −10.59620483461028847626822602166, −9.624655657930705674949582588673, −8.788433576970449811671959604984, −6.54067366549527496996977554374, −4.63651764185132023479331374934, −2.23059667538637953409462863522, 1.20443045161012741216697131796, 4.13419050154590524545291388731, 6.50998590669358350362979104618, 8.197424160893559208416891909850, 8.605726874918311116770106784981, 10.41161229037126117570235425245, 12.26229018820420607009256442629, 13.45338858380829925206629189127, 13.92392785426157350011241864739, 16.09316636374426924533042161892

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