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Properties

Label 2-70-5.4-c1-0-1
Degree $2$
Conductor $70$
Sign $0.994 - 0.100i$
Analytic cond. $0.558952$
Root an. cond. $0.747631$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − i·2-s + 2.44i·3-s − 4-s + (2.22 − 0.224i)5-s + 2.44·6-s − i·7-s + i·8-s − 2.99·9-s + (−0.224 − 2.22i)10-s − 4.89·11-s − 2.44i·12-s − 4.44i·13-s − 14-s + (0.550 + 5.44i)15-s + 16-s + 2i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + 1.41i·3-s − 0.5·4-s + (0.994 − 0.100i)5-s + 0.999·6-s − 0.377i·7-s + 0.353i·8-s − 0.999·9-s + (−0.0710 − 0.703i)10-s − 1.47·11-s − 0.707i·12-s − 1.23i·13-s − 0.267·14-s + (0.142 + 1.40i)15-s + 0.250·16-s + 0.485i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.100i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 - 0.100i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(70\)    =    \(2 \cdot 5 \cdot 7\)
Sign: $0.994 - 0.100i$
Analytic conductor: \(0.558952\)
Root analytic conductor: \(0.747631\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{70} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 70,\ (\ :1/2),\ 0.994 - 0.100i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.942687 + 0.0474944i\)
\(L(\frac12)\) \(\approx\) \(0.942687 + 0.0474944i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
5 \( 1 + (-2.22 + 0.224i)T \)
7 \( 1 + iT \)
good3 \( 1 - 2.44iT - 3T^{2} \)
11 \( 1 + 4.89T + 11T^{2} \)
13 \( 1 + 4.44iT - 13T^{2} \)
17 \( 1 - 2iT - 17T^{2} \)
19 \( 1 + 1.55T + 19T^{2} \)
23 \( 1 + 2.89iT - 23T^{2} \)
29 \( 1 + 6.89T + 29T^{2} \)
31 \( 1 - 8.89T + 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 + 1.10T + 41T^{2} \)
43 \( 1 - 0.898iT - 43T^{2} \)
47 \( 1 - 8.89iT - 47T^{2} \)
53 \( 1 - 10.8iT - 53T^{2} \)
59 \( 1 - 1.55T + 59T^{2} \)
61 \( 1 - 3.55T + 61T^{2} \)
67 \( 1 + 8iT - 67T^{2} \)
71 \( 1 + 1.10T + 71T^{2} \)
73 \( 1 + 2.89iT - 73T^{2} \)
79 \( 1 + 6.89T + 79T^{2} \)
83 \( 1 - 2.44iT - 83T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 - 15.7iT - 97T^{2} \)
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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

−14.80967854907105242526667714811, −13.50002272829615240033097414624, −12.69730178009787096880303935153, −10.73426123510157600035228413424, −10.43134582684158402341125059784, −9.522611812166983285514215208849, −8.163324242375110243183970953295, −5.69448156402581075408846634244, −4.61801799198574011265464084909, −2.90486681587310966934221475522, 2.20761612620777380177549525069, 5.29435469974098243385809056628, 6.44253967075169279419816373286, 7.41215864755701840767877718374, 8.648810880927592417393614859814, 9.972584520176235140764895497419, 11.66693394927099345644334706199, 13.01986328622609529127605007166, 13.45973863280713756712153726458, 14.44646370975234255480349180264

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