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Properties

Label 2-70-5.4-c3-0-2
Degree $2$
Conductor $70$
Sign $-0.894 - 0.447i$
Analytic cond. $4.13013$
Root an. cond. $2.03227$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2i·2-s + 7i·3-s − 4·4-s + (10 + 5i)5-s − 14·6-s + 7i·7-s − 8i·8-s − 22·9-s + (−10 + 20i)10-s − 37·11-s − 28i·12-s − 51i·13-s − 14·14-s + (−35 + 70i)15-s + 16·16-s + 41i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + 1.34i·3-s − 0.5·4-s + (0.894 + 0.447i)5-s − 0.952·6-s + 0.377i·7-s − 0.353i·8-s − 0.814·9-s + (−0.316 + 0.632i)10-s − 1.01·11-s − 0.673i·12-s − 1.08i·13-s − 0.267·14-s + (−0.602 + 1.20i)15-s + 0.250·16-s + 0.584i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(70\)    =    \(2 \cdot 5 \cdot 7\)
Sign: $-0.894 - 0.447i$
Analytic conductor: \(4.13013\)
Root analytic conductor: \(2.03227\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{70} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 70,\ (\ :3/2),\ -0.894 - 0.447i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.338622 + 1.43442i\)
\(L(\frac12)\) \(\approx\) \(0.338622 + 1.43442i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2iT \)
5 \( 1 + (-10 - 5i)T \)
7 \( 1 - 7iT \)
good3 \( 1 - 7iT - 27T^{2} \)
11 \( 1 + 37T + 1.33e3T^{2} \)
13 \( 1 + 51iT - 2.19e3T^{2} \)
17 \( 1 - 41iT - 4.91e3T^{2} \)
19 \( 1 - 108T + 6.85e3T^{2} \)
23 \( 1 - 70iT - 1.21e4T^{2} \)
29 \( 1 - 249T + 2.43e4T^{2} \)
31 \( 1 + 134T + 2.97e4T^{2} \)
37 \( 1 + 334iT - 5.06e4T^{2} \)
41 \( 1 - 206T + 6.89e4T^{2} \)
43 \( 1 - 376iT - 7.95e4T^{2} \)
47 \( 1 + 287iT - 1.03e5T^{2} \)
53 \( 1 - 6iT - 1.48e5T^{2} \)
59 \( 1 - 2T + 2.05e5T^{2} \)
61 \( 1 + 940T + 2.26e5T^{2} \)
67 \( 1 - 106iT - 3.00e5T^{2} \)
71 \( 1 - 456T + 3.57e5T^{2} \)
73 \( 1 + 650iT - 3.89e5T^{2} \)
79 \( 1 - 1.23e3T + 4.93e5T^{2} \)
83 \( 1 + 428iT - 5.71e5T^{2} \)
89 \( 1 - 220T + 7.04e5T^{2} \)
97 \( 1 + 1.05e3iT - 9.12e5T^{2} \)
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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

−14.95874643872354539388060303324, −13.90514877564448857311366002736, −12.75209081584646640439851691702, −10.82667295576835483070048417340, −10.06365754421865754019791259563, −9.158416182249837633016151765760, −7.68636113125550124844878375250, −5.85196397332882706266720027416, −5.05328577804960084359990535266, −3.16164617336304600474876871373, 1.08181718597291226594940416913, 2.49723879914460301259313473329, 4.96533861786912371008243909024, 6.53912637881253195178942533153, 7.83490811575240044174261710539, 9.209710868199050433289692142871, 10.39445098877789195841852273401, 11.83451835575880977850606259165, 12.66087495841957018935546270111, 13.69891548369657476228828402675

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