Properties

Label 2-147-7.2-c7-0-29
Degree $2$
Conductor $147$
Sign $0.386 + 0.922i$
Analytic cond. $45.9205$
Root an. cond. $6.77647$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (6.10 − 10.5i)2-s + (−13.5 − 23.3i)3-s + (−10.5 − 18.2i)4-s + (142. − 246. i)5-s − 329.·6-s + 1.30e3·8-s + (−364.5 + 631. i)9-s + (−1.73e3 − 3.01e3i)10-s + (3.64e3 + 6.31e3i)11-s + (−285. + 493. i)12-s + 8.51e3·13-s − 7.68e3·15-s + (9.32e3 − 1.61e4i)16-s + (1.67e4 + 2.89e4i)17-s + (4.45e3 + 7.70e3i)18-s + (4.45e3 − 7.71e3i)19-s + ⋯
L(s)  = 1  + (0.539 − 0.934i)2-s + (−0.288 − 0.499i)3-s + (−0.0825 − 0.142i)4-s + (0.509 − 0.882i)5-s − 0.623·6-s + 0.901·8-s + (−0.166 + 0.288i)9-s + (−0.549 − 0.952i)10-s + (0.825 + 1.42i)11-s + (−0.0476 + 0.0825i)12-s + 1.07·13-s − 0.588·15-s + (0.568 − 0.985i)16-s + (0.825 + 1.43i)17-s + (0.179 + 0.311i)18-s + (0.148 − 0.257i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $0.386 + 0.922i$
Analytic conductor: \(45.9205\)
Root analytic conductor: \(6.77647\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :7/2),\ 0.386 + 0.922i)\)

Particular Values

\(L(4)\) \(\approx\) \(3.586381336\)
\(L(\frac12)\) \(\approx\) \(3.586381336\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (13.5 + 23.3i)T \)
7 \( 1 \)
good2 \( 1 + (-6.10 + 10.5i)T + (-64 - 110. i)T^{2} \)
5 \( 1 + (-142. + 246. i)T + (-3.90e4 - 6.76e4i)T^{2} \)
11 \( 1 + (-3.64e3 - 6.31e3i)T + (-9.74e6 + 1.68e7i)T^{2} \)
13 \( 1 - 8.51e3T + 6.27e7T^{2} \)
17 \( 1 + (-1.67e4 - 2.89e4i)T + (-2.05e8 + 3.55e8i)T^{2} \)
19 \( 1 + (-4.45e3 + 7.71e3i)T + (-4.46e8 - 7.74e8i)T^{2} \)
23 \( 1 + (2.72e4 - 4.71e4i)T + (-1.70e9 - 2.94e9i)T^{2} \)
29 \( 1 + 3.24e4T + 1.72e10T^{2} \)
31 \( 1 + (-8.45e4 - 1.46e5i)T + (-1.37e10 + 2.38e10i)T^{2} \)
37 \( 1 + (4.96e4 - 8.59e4i)T + (-4.74e10 - 8.22e10i)T^{2} \)
41 \( 1 - 2.35e5T + 1.94e11T^{2} \)
43 \( 1 + 2.89e5T + 2.71e11T^{2} \)
47 \( 1 + (-6.04e5 + 1.04e6i)T + (-2.53e11 - 4.38e11i)T^{2} \)
53 \( 1 + (-9.89e4 - 1.71e5i)T + (-5.87e11 + 1.01e12i)T^{2} \)
59 \( 1 + (1.11e6 + 1.92e6i)T + (-1.24e12 + 2.15e12i)T^{2} \)
61 \( 1 + (5.30e5 - 9.18e5i)T + (-1.57e12 - 2.72e12i)T^{2} \)
67 \( 1 + (1.87e6 + 3.24e6i)T + (-3.03e12 + 5.24e12i)T^{2} \)
71 \( 1 + 2.99e6T + 9.09e12T^{2} \)
73 \( 1 + (3.28e6 + 5.69e6i)T + (-5.52e12 + 9.56e12i)T^{2} \)
79 \( 1 + (3.54e6 - 6.14e6i)T + (-9.60e12 - 1.66e13i)T^{2} \)
83 \( 1 + 4.41e5T + 2.71e13T^{2} \)
89 \( 1 + (5.92e6 - 1.02e7i)T + (-2.21e13 - 3.83e13i)T^{2} \)
97 \( 1 - 4.49e6T + 8.07e13T^{2} \)
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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

−11.92695124740729032605461515622, −10.72758016254858843148500726313, −9.729485765703523612443861422641, −8.469929457853419244997646792815, −7.23010163012633003884373828935, −5.89404386556545927708035907722, −4.66742180985413836137642096944, −3.56011387611497406119922586134, −1.72372403479761316494863089348, −1.35619670645607308027649283627, 0.998571549406775535855489345688, 3.01464677831036009769022629561, 4.28430354580104729973503262554, 5.87706382077652038011646073457, 6.09230369089135261145654767301, 7.33489609802139204072823477900, 8.715117541148683485467210163993, 10.01117563979775456203459955483, 10.89741843927464523439439492620, 11.69912600178178114351582622306

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