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Properties

Label 2-70-35.18-c2-0-0
Degree $2$
Conductor $70$
Sign $-0.644 - 0.764i$
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 + (−5.10 + 1.36i)3-s + (1.73 + i)4-s + (−1.96 + 4.59i)5-s − 7.47·6-s + (−6.41 + 2.80i)7-s + (1.99 + 2i)8-s + (16.4 − 9.47i)9-s + (−4.36 + 5.56i)10-s + (−1.73 + 3.01i)11-s + (−10.2 − 2.73i)12-s + (10.9 + 10.9i)13-s + (−9.78 + 1.47i)14-s + (3.73 − 26.1i)15-s + (1.99 + 3.46i)16-s + (−2.54 − 9.49i)17-s + ⋯
L(s)  = 1  + (0.683 + 0.183i)2-s + (−1.70 + 0.456i)3-s + (0.433 + 0.250i)4-s + (−0.392 + 0.919i)5-s − 1.24·6-s + (−0.916 + 0.400i)7-s + (0.249 + 0.250i)8-s + (1.82 − 1.05i)9-s + (−0.436 + 0.556i)10-s + (−0.158 + 0.273i)11-s + (−0.851 − 0.228i)12-s + (0.842 + 0.842i)13-s + (−0.699 + 0.105i)14-s + (0.249 − 1.74i)15-s + (0.124 + 0.216i)16-s + (−0.149 − 0.558i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.644 - 0.764i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.644 - 0.764i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(70\)    =    \(2 \cdot 5 \cdot 7\)
Sign: $-0.644 - 0.764i$
Analytic conductor: \(1.90736\)
Root analytic conductor: \(1.38107\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{70} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 70,\ (\ :1),\ -0.644 - 0.764i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.346080 + 0.743919i\)
\(L(\frac12)\) \(\approx\) \(0.346080 + 0.743919i\)
\(L(2)\) 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 \( 1 + (6.41 - 2.80i)T \)
good3 \( 1 + (5.10 - 1.36i)T + (7.79 - 4.5i)T^{2} \)
11 \( 1 + (1.73 - 3.01i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (-10.9 - 10.9i)T + 169iT^{2} \)
17 \( 1 + (2.54 + 9.49i)T + (-250. + 144.5i)T^{2} \)
19 \( 1 + (-15.9 + 9.21i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (4.64 - 17.3i)T + (-458. - 264.5i)T^{2} \)
29 \( 1 - 49.8iT - 841T^{2} \)
31 \( 1 + (8 - 13.8i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (-34.2 - 9.16i)T + (1.18e3 + 684.5i)T^{2} \)
41 \( 1 + 6.04T + 1.68e3T^{2} \)
43 \( 1 + (9.64 + 9.64i)T + 1.84e3iT^{2} \)
47 \( 1 + (20.3 + 5.44i)T + (1.91e3 + 1.10e3i)T^{2} \)
53 \( 1 + (-10.9 + 2.94i)T + (2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (-65.3 - 37.7i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (51.4 + 89.1i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (8.33 + 31.1i)T + (-3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 - 40.3T + 5.04e3T^{2} \)
73 \( 1 + (-85.7 + 22.9i)T + (4.61e3 - 2.66e3i)T^{2} \)
79 \( 1 + (-87.6 + 50.6i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-51.1 - 51.1i)T + 6.88e3iT^{2} \)
89 \( 1 + (71.6 - 41.3i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-45.7 + 45.7i)T - 9.40e3iT^{2} \)
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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

−15.18085707329436084179514678990, −13.69045125919544770859895652127, −12.38466225653843643190713837271, −11.54516519336027384095024019374, −10.81720255611793575213223692983, −9.550253823048233940851589460999, −7.05431888732550479459770588833, −6.34668454974522843695412420526, −5.12132010678239212373124961946, −3.55668901443372245323589294887, 0.73756352145498310504947208972, 4.06551705740924862542412377917, 5.54202476405790014026715186385, 6.31720730031604419424802306126, 7.84694677599315320243156362139, 9.981597865301497605769947742289, 11.06880734278230096222867378376, 12.00753600113023323191495236438, 12.88605638794226141721781754304, 13.40535885205262125836358771150

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