Properties

Label 2-70-35.2-c2-0-6
Degree $2$
Conductor $70$
Sign $0.967 + 0.254i$
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  + (1.36 − 0.366i)2-s + (2.37 + 0.636i)3-s + (1.73 − i)4-s + (−1.96 − 4.59i)5-s + 3.47·6-s + (0.219 + 6.99i)7-s + (1.99 − 2i)8-s + (−2.55 − 1.47i)9-s + (−4.36 − 5.56i)10-s + (3.73 + 6.47i)11-s + (4.74 − 1.27i)12-s + (−10.9 + 10.9i)13-s + (2.86 + 9.47i)14-s + (−1.73 − 12.1i)15-s + (1.99 − 3.46i)16-s + (5.47 − 20.4i)17-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s + (0.791 + 0.212i)3-s + (0.433 − 0.250i)4-s + (−0.392 − 0.919i)5-s + 0.579·6-s + (0.0313 + 0.999i)7-s + (0.249 − 0.250i)8-s + (−0.284 − 0.164i)9-s + (−0.436 − 0.556i)10-s + (0.339 + 0.588i)11-s + (0.395 − 0.106i)12-s + (−0.842 + 0.842i)13-s + (0.204 + 0.676i)14-s + (−0.115 − 0.811i)15-s + (0.124 − 0.216i)16-s + (0.321 − 1.20i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 + 0.254i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.967 + 0.254i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(70\)    =    \(2 \cdot 5 \cdot 7\)
Sign: $0.967 + 0.254i$
Analytic conductor: \(1.90736\)
Root analytic conductor: \(1.38107\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{70} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 70,\ (\ :1),\ 0.967 + 0.254i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.95952 - 0.253215i\)
\(L(\frac12)\) \(\approx\) \(1.95952 - 0.253215i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.36 + 0.366i)T \)
5 \( 1 + (1.96 + 4.59i)T \)
7 \( 1 + (-0.219 - 6.99i)T \)
good3 \( 1 + (-2.37 - 0.636i)T + (7.79 + 4.5i)T^{2} \)
11 \( 1 + (-3.73 - 6.47i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (10.9 - 10.9i)T - 169iT^{2} \)
17 \( 1 + (-5.47 + 20.4i)T + (-250. - 144.5i)T^{2} \)
19 \( 1 + (12.4 + 7.21i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-5.37 - 20.0i)T + (-458. + 264.5i)T^{2} \)
29 \( 1 - 15.8iT - 841T^{2} \)
31 \( 1 + (8 + 13.8i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (-64.1 + 17.1i)T + (1.18e3 - 684.5i)T^{2} \)
41 \( 1 + 27.9T + 1.68e3T^{2} \)
43 \( 1 + (-39.6 + 39.6i)T - 1.84e3iT^{2} \)
47 \( 1 + (-69.4 + 18.6i)T + (1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (-40.9 - 10.9i)T + (2.43e3 + 1.40e3i)T^{2} \)
59 \( 1 + (-55.8 + 32.2i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (29.5 - 51.1i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (14.3 - 53.5i)T + (-3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + 36.3T + 5.04e3T^{2} \)
73 \( 1 + (63.8 + 17.1i)T + (4.61e3 + 2.66e3i)T^{2} \)
79 \( 1 + (35.6 + 20.6i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (9.12 - 9.12i)T - 6.88e3iT^{2} \)
89 \( 1 + (-42.1 - 24.3i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (63.7 + 63.7i)T + 9.40e3iT^{2} \)
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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

−14.53995446620784808498233352903, −13.34715100571235485741704180963, −12.14816390220084554637593637080, −11.62415208889764957583170763149, −9.492402002557923371409808442532, −8.916103846894158305778682687759, −7.36053209676002492238613256394, −5.48347412361007215657862965310, −4.21085560513011392394793889662, −2.45842477872719908388430326072, 2.80748562330978601404765767078, 4.05273033824727681881922385953, 6.12592323483649416212848759168, 7.47376506332231228572742695614, 8.265683414298484468019511794748, 10.27295809001553888493006023144, 11.13467044479855519034060455685, 12.59988705652806611402713321839, 13.65583954213048441817352478553, 14.60670850318483224947902148294

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