Properties

Label 2-117-117.110-c1-0-1
Degree $2$
Conductor $117$
Sign $-0.999 - 0.0171i$
Analytic cond. $0.934249$
Root an. cond. $0.966565$
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  + (−0.641 + 2.39i)2-s + (0.658 + 1.60i)3-s + (−3.59 − 2.07i)4-s + (−0.145 + 0.542i)5-s + (−4.25 + 0.550i)6-s + (−0.396 − 0.396i)7-s + (3.76 − 3.76i)8-s + (−2.13 + 2.11i)9-s + (−1.20 − 0.695i)10-s + (5.09 + 1.36i)11-s + (0.954 − 7.11i)12-s + (−1.60 − 3.22i)13-s + (1.20 − 0.694i)14-s + (−0.964 + 0.124i)15-s + (2.44 + 4.24i)16-s + (1.39 + 2.42i)17-s + ⋯
L(s)  = 1  + (−0.453 + 1.69i)2-s + (0.380 + 0.924i)3-s + (−1.79 − 1.03i)4-s + (−0.0649 + 0.242i)5-s + (−1.73 + 0.224i)6-s + (−0.149 − 0.149i)7-s + (1.33 − 1.33i)8-s + (−0.710 + 0.703i)9-s + (−0.381 − 0.220i)10-s + (1.53 + 0.411i)11-s + (0.275 − 2.05i)12-s + (−0.446 − 0.894i)13-s + (0.321 − 0.185i)14-s + (−0.248 + 0.0321i)15-s + (0.612 + 1.06i)16-s + (0.339 + 0.587i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0171i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0171i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $-0.999 - 0.0171i$
Analytic conductor: \(0.934249\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (110, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1/2),\ -0.999 - 0.0171i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00729683 + 0.849126i\)
\(L(\frac12)\) \(\approx\) \(0.00729683 + 0.849126i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.658 - 1.60i)T \)
13 \( 1 + (1.60 + 3.22i)T \)
good2 \( 1 + (0.641 - 2.39i)T + (-1.73 - i)T^{2} \)
5 \( 1 + (0.145 - 0.542i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (0.396 + 0.396i)T + 7iT^{2} \)
11 \( 1 + (-5.09 - 1.36i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-1.39 - 2.42i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.31 - 0.888i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + 7.55T + 23T^{2} \)
29 \( 1 + (0.775 - 0.447i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-5.88 - 1.57i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (0.0565 - 0.0151i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-4.11 - 4.11i)T + 41iT^{2} \)
43 \( 1 + 10.4iT - 43T^{2} \)
47 \( 1 + (2.84 + 10.6i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 - 5.68iT - 53T^{2} \)
59 \( 1 + (1.32 + 4.93i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + 4.16T + 61T^{2} \)
67 \( 1 + (-4.65 + 4.65i)T - 67iT^{2} \)
71 \( 1 + (-0.573 + 2.14i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-2.68 - 2.68i)T + 73iT^{2} \)
79 \( 1 + (-2.61 + 4.52i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.139 + 0.0375i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (2.14 + 7.99i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (0.650 - 0.650i)T - 97iT^{2} \)
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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

−14.50448259198277493664390653945, −13.72798790549804162725906961993, −11.99453459447012580164818203448, −10.29275881308558758089496533815, −9.593339356501593861013648403950, −8.566352101358049246855034948202, −7.59545168503668040737296928738, −6.37447555426665871165254067029, −5.18029122454891320131837734598, −3.79673713510500950912822940022, 1.23553772408218504883518935451, 2.75996650092368901323036300243, 4.16879068963438969427127713689, 6.42828934935576433736793819640, 7.982593043848026560369730064139, 9.129313814532336549533359727826, 9.647523757034689198247905885418, 11.36558895314073002018841525286, 11.92614819243457866688741396746, 12.57163449497007843846610600094

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