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Properties

Label 2-70-35.23-c2-0-3
Degree $2$
Conductor $70$
Sign $0.468 - 0.883i$
Analytic cond. $1.90736$
Root an. cond. $1.38107$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.366 + 1.36i)2-s + (0.303 − 1.13i)3-s + (−1.73 + i)4-s + (2.53 + 4.30i)5-s + 1.65·6-s + (4.91 + 4.97i)7-s + (−2 − 1.99i)8-s + (6.60 + 3.81i)9-s + (−4.95 + 5.04i)10-s + (−6.06 − 10.5i)11-s + (0.606 + 2.26i)12-s + (−8.70 − 8.70i)13-s + (−5.00 + 8.54i)14-s + (5.64 − 1.56i)15-s + (1.99 − 3.46i)16-s + (5.40 + 1.44i)17-s + ⋯
L(s)  = 1  + (0.183 + 0.683i)2-s + (0.101 − 0.376i)3-s + (−0.433 + 0.250i)4-s + (0.507 + 0.861i)5-s + 0.275·6-s + (0.702 + 0.711i)7-s + (−0.250 − 0.249i)8-s + (0.734 + 0.423i)9-s + (−0.495 + 0.504i)10-s + (−0.551 − 0.955i)11-s + (0.0505 + 0.188i)12-s + (−0.669 − 0.669i)13-s + (−0.357 + 0.610i)14-s + (0.376 − 0.104i)15-s + (0.124 − 0.216i)16-s + (0.318 + 0.0852i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.468 - 0.883i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.468 - 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(70\)    =    \(2 \cdot 5 \cdot 7\)
Sign: $0.468 - 0.883i$
Analytic conductor: \(1.90736\)
Root analytic conductor: \(1.38107\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{70} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 70,\ (\ :1),\ 0.468 - 0.883i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.24906 + 0.751263i\)
\(L(\frac12)\) \(\approx\) \(1.24906 + 0.751263i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.366 - 1.36i)T \)
5 \( 1 + (-2.53 - 4.30i)T \)
7 \( 1 + (-4.91 - 4.97i)T \)
good3 \( 1 + (-0.303 + 1.13i)T + (-7.79 - 4.5i)T^{2} \)
11 \( 1 + (6.06 + 10.5i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (8.70 + 8.70i)T + 169iT^{2} \)
17 \( 1 + (-5.40 - 1.44i)T + (250. + 144.5i)T^{2} \)
19 \( 1 + (26.9 + 15.5i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-30.1 + 8.07i)T + (458. - 264.5i)T^{2} \)
29 \( 1 + 25.2iT - 841T^{2} \)
31 \( 1 + (13.4 + 23.2i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (-14.8 - 55.5i)T + (-1.18e3 + 684.5i)T^{2} \)
41 \( 1 + 45.8T + 1.68e3T^{2} \)
43 \( 1 + (-18.2 - 18.2i)T + 1.84e3iT^{2} \)
47 \( 1 + (1.44 + 5.40i)T + (-1.91e3 + 1.10e3i)T^{2} \)
53 \( 1 + (-8.58 + 32.0i)T + (-2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (-51.0 + 29.4i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (37.7 - 65.4i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (54.1 + 14.5i)T + (3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 - 22.2T + 5.04e3T^{2} \)
73 \( 1 + (0.862 - 3.21i)T + (-4.61e3 - 2.66e3i)T^{2} \)
79 \( 1 + (37.4 + 21.6i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (35.1 + 35.1i)T + 6.88e3iT^{2} \)
89 \( 1 + (-11.8 - 6.85i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (58.3 - 58.3i)T - 9.40e3iT^{2} \)
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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

−14.89047156093729992835136285209, −13.56512234411729823053990590466, −12.87862147664972816299156133601, −11.27707311929675286762806742956, −10.15725508140662548967598394738, −8.561021427444442146673054921995, −7.51442100514440791188128730709, −6.25888345728545446995519098971, −4.97317964933103734020610137118, −2.61028469622406981627287001731, 1.69713470197546054190739698429, 4.18875823513730572665482285070, 5.07336822832277349060876175096, 7.20268372686599546178019710670, 8.841477441772690515159997063115, 9.890239198293025558363099697759, 10.72945987167922210116094196694, 12.33807272978227258078079147170, 12.90400461042389199433757075746, 14.22911028264471272168289920130

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