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Properties

Label 2-71-71.25-c1-0-0
Degree $2$
Conductor $71$
Sign $-0.779 - 0.626i$
Analytic cond. $0.566937$
Root an. cond. $0.752952$
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  + (−1.27 + 0.924i)2-s + (−0.850 + 0.617i)3-s + (0.146 − 0.452i)4-s + (0.148 + 0.456i)5-s + (0.511 − 1.57i)6-s + (−4.12 + 2.99i)7-s + (−0.741 − 2.28i)8-s + (−0.585 + 1.80i)9-s + (−0.610 − 0.443i)10-s + (3.85 − 2.80i)11-s + (0.154 + 0.475i)12-s + (3.98 + 2.89i)13-s + (2.47 − 7.62i)14-s + (−0.408 − 0.296i)15-s + (3.82 + 2.77i)16-s + (1.54 + 1.12i)17-s + ⋯
L(s)  = 1  + (−0.900 + 0.653i)2-s + (−0.490 + 0.356i)3-s + (0.0734 − 0.226i)4-s + (0.0663 + 0.204i)5-s + (0.208 − 0.642i)6-s + (−1.55 + 1.13i)7-s + (−0.262 − 0.806i)8-s + (−0.195 + 0.600i)9-s + (−0.193 − 0.140i)10-s + (1.16 − 0.845i)11-s + (0.0445 + 0.137i)12-s + (1.10 + 0.803i)13-s + (0.662 − 2.03i)14-s + (−0.105 − 0.0765i)15-s + (0.955 + 0.694i)16-s + (0.375 + 0.272i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.779 - 0.626i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.779 - 0.626i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(71\)
Sign: $-0.779 - 0.626i$
Analytic conductor: \(0.566937\)
Root analytic conductor: \(0.752952\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{71} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 71,\ (\ :1/2),\ -0.779 - 0.626i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.142502 + 0.404464i\)
\(L(\frac12)\) \(\approx\) \(0.142502 + 0.404464i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad71 \( 1 + (2.21 + 8.12i)T \)
good2 \( 1 + (1.27 - 0.924i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (0.850 - 0.617i)T + (0.927 - 2.85i)T^{2} \)
5 \( 1 + (-0.148 - 0.456i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (4.12 - 2.99i)T + (2.16 - 6.65i)T^{2} \)
11 \( 1 + (-3.85 + 2.80i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (-3.98 - 2.89i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-1.54 - 1.12i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.01 - 0.734i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 2.45T + 23T^{2} \)
29 \( 1 + (-1.32 - 4.06i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (1.22 - 0.892i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + 5.25T + 37T^{2} \)
41 \( 1 + 1.67T + 41T^{2} \)
43 \( 1 + (-1.95 - 6.02i)T + (-34.7 + 25.2i)T^{2} \)
47 \( 1 + (-6.28 - 4.56i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (2.81 + 8.67i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (0.683 + 2.10i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-7.83 - 5.68i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (0.465 - 1.43i)T + (-54.2 - 39.3i)T^{2} \)
73 \( 1 + (-4.29 + 3.12i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-0.954 - 2.93i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (0.659 + 2.02i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (0.424 + 1.30i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 - 4.29T + 97T^{2} \)
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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

−15.73804493937862918073797147252, −14.19764580751315569395819658158, −12.83062484280975422782407277771, −11.72963522451268633543881437912, −10.35599536257931374131445948947, −9.168875511673005211338449961171, −8.533667355120519039701127777981, −6.57920578233947738628397518291, −6.01074938431910856881444701586, −3.53513294087320603580236648297, 0.870737830100476689953827958645, 3.60448417907229069849671497084, 6.00105151743203618963098359604, 7.08752954162666632325841511019, 8.917574023824277933132686968338, 9.800002610393990022808621180891, 10.69167941527025731087248358338, 11.95116810150928417153799147895, 12.86727350007402101760766876485, 14.06991982804718653699929897131

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