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Properties

Label 2-112-7.2-c7-0-15
Degree $2$
Conductor $112$
Sign $-0.793 + 0.608i$
Analytic cond. $34.9871$
Root an. cond. $5.91499$
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  + (−29.3 − 50.8i)3-s + (−209. + 362. i)5-s + (505. + 753. i)7-s + (−631. + 1.09e3i)9-s + (128. + 222. i)11-s − 8.02e3·13-s + 2.45e4·15-s + (1.28e4 + 2.22e4i)17-s + (2.75e4 − 4.77e4i)19-s + (2.34e4 − 4.78e4i)21-s + (−1.89e4 + 3.28e4i)23-s + (−4.84e4 − 8.39e4i)25-s − 5.42e4·27-s + 1.25e5·29-s + (−8.01e4 − 1.38e5i)31-s + ⋯
L(s)  = 1  + (−0.628 − 1.08i)3-s + (−0.748 + 1.29i)5-s + (0.557 + 0.830i)7-s + (−0.288 + 0.500i)9-s + (0.0290 + 0.0503i)11-s − 1.01·13-s + 1.88·15-s + (0.632 + 1.09i)17-s + (0.921 − 1.59i)19-s + (0.552 − 1.12i)21-s + (−0.324 + 0.562i)23-s + (−0.620 − 1.07i)25-s − 0.530·27-s + 0.957·29-s + (−0.483 − 0.837i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.793 + 0.608i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.793 + 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $-0.793 + 0.608i$
Analytic conductor: \(34.9871\)
Root analytic conductor: \(5.91499\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{112} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 112,\ (\ :7/2),\ -0.793 + 0.608i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.3641469953\)
\(L(\frac12)\) \(\approx\) \(0.3641469953\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (-505. - 753. i)T \)
good3 \( 1 + (29.3 + 50.8i)T + (-1.09e3 + 1.89e3i)T^{2} \)
5 \( 1 + (209. - 362. i)T + (-3.90e4 - 6.76e4i)T^{2} \)
11 \( 1 + (-128. - 222. i)T + (-9.74e6 + 1.68e7i)T^{2} \)
13 \( 1 + 8.02e3T + 6.27e7T^{2} \)
17 \( 1 + (-1.28e4 - 2.22e4i)T + (-2.05e8 + 3.55e8i)T^{2} \)
19 \( 1 + (-2.75e4 + 4.77e4i)T + (-4.46e8 - 7.74e8i)T^{2} \)
23 \( 1 + (1.89e4 - 3.28e4i)T + (-1.70e9 - 2.94e9i)T^{2} \)
29 \( 1 - 1.25e5T + 1.72e10T^{2} \)
31 \( 1 + (8.01e4 + 1.38e5i)T + (-1.37e10 + 2.38e10i)T^{2} \)
37 \( 1 + (7.05e4 - 1.22e5i)T + (-4.74e10 - 8.22e10i)T^{2} \)
41 \( 1 + 6.74e5T + 1.94e11T^{2} \)
43 \( 1 + 7.55e5T + 2.71e11T^{2} \)
47 \( 1 + (-1.02e5 + 1.77e5i)T + (-2.53e11 - 4.38e11i)T^{2} \)
53 \( 1 + (1.82e5 + 3.16e5i)T + (-5.87e11 + 1.01e12i)T^{2} \)
59 \( 1 + (9.62e5 + 1.66e6i)T + (-1.24e12 + 2.15e12i)T^{2} \)
61 \( 1 + (-1.35e6 + 2.35e6i)T + (-1.57e12 - 2.72e12i)T^{2} \)
67 \( 1 + (7.82e5 + 1.35e6i)T + (-3.03e12 + 5.24e12i)T^{2} \)
71 \( 1 - 2.29e6T + 9.09e12T^{2} \)
73 \( 1 + (-5.86e5 - 1.01e6i)T + (-5.52e12 + 9.56e12i)T^{2} \)
79 \( 1 + (-3.60e5 + 6.23e5i)T + (-9.60e12 - 1.66e13i)T^{2} \)
83 \( 1 + 2.39e6T + 2.71e13T^{2} \)
89 \( 1 + (-6.20e6 + 1.07e7i)T + (-2.21e13 - 3.83e13i)T^{2} \)
97 \( 1 + 1.48e7T + 8.07e13T^{2} \)
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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

−11.73564415510021141531098014378, −11.26369963796173827940073026683, −9.867687408672285169289898801213, −8.152896221236364715477201490492, −7.26712198298389989743675000943, −6.46527201913757902540499725823, −5.13665034972510630861714462425, −3.20426874986922163275680042430, −1.87003223021953067924091291535, −0.13398468200975692936126110609, 1.13911485630111315178738894960, 3.70075060836182266013730222143, 4.75744391689488749798818379121, 5.27126450823334650247892009029, 7.37406325922400217079484553430, 8.352398738613816440753789316558, 9.726053006008497634470205543309, 10.42552408005258549685646688368, 11.80326470874091797820712192519, 12.19074497175497856758105250504

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