Properties

Label 2-112-7.4-c7-0-11
Degree $2$
Conductor $112$
Sign $-0.113 - 0.993i$
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  + (−1.40 + 2.42i)3-s + (219. + 379. i)5-s + (893. − 159. i)7-s + (1.08e3 + 1.88e3i)9-s + (−2.74e3 + 4.74e3i)11-s + 4.00e3·13-s − 1.22e3·15-s + (1.40e4 − 2.42e4i)17-s + (1.19e4 + 2.06e4i)19-s + (−864. + 2.39e3i)21-s + (−3.68e4 − 6.38e4i)23-s + (−5.69e4 + 9.86e4i)25-s − 1.22e4·27-s − 9.87e4·29-s + (−2.37e4 + 4.11e4i)31-s + ⋯
L(s)  = 1  + (−0.0299 + 0.0519i)3-s + (0.784 + 1.35i)5-s + (0.984 − 0.176i)7-s + (0.498 + 0.862i)9-s + (−0.620 + 1.07i)11-s + 0.505·13-s − 0.0940·15-s + (0.691 − 1.19i)17-s + (0.398 + 0.690i)19-s + (−0.0203 + 0.0564i)21-s + (−0.631 − 1.09i)23-s + (−0.729 + 1.26i)25-s − 0.119·27-s − 0.751·29-s + (−0.143 + 0.247i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.113 - 0.993i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.113 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $-0.113 - 0.993i$
Analytic conductor: \(34.9871\)
Root analytic conductor: \(5.91499\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{112} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 112,\ (\ :7/2),\ -0.113 - 0.993i)\)

Particular Values

\(L(4)\) \(\approx\) \(2.581676970\)
\(L(\frac12)\) \(\approx\) \(2.581676970\)
\(L(\frac{9}{2})\) 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 \( 1 + (-893. + 159. i)T \)
good3 \( 1 + (1.40 - 2.42i)T + (-1.09e3 - 1.89e3i)T^{2} \)
5 \( 1 + (-219. - 379. i)T + (-3.90e4 + 6.76e4i)T^{2} \)
11 \( 1 + (2.74e3 - 4.74e3i)T + (-9.74e6 - 1.68e7i)T^{2} \)
13 \( 1 - 4.00e3T + 6.27e7T^{2} \)
17 \( 1 + (-1.40e4 + 2.42e4i)T + (-2.05e8 - 3.55e8i)T^{2} \)
19 \( 1 + (-1.19e4 - 2.06e4i)T + (-4.46e8 + 7.74e8i)T^{2} \)
23 \( 1 + (3.68e4 + 6.38e4i)T + (-1.70e9 + 2.94e9i)T^{2} \)
29 \( 1 + 9.87e4T + 1.72e10T^{2} \)
31 \( 1 + (2.37e4 - 4.11e4i)T + (-1.37e10 - 2.38e10i)T^{2} \)
37 \( 1 + (-5.00e4 - 8.66e4i)T + (-4.74e10 + 8.22e10i)T^{2} \)
41 \( 1 - 4.89e5T + 1.94e11T^{2} \)
43 \( 1 + 2.99e5T + 2.71e11T^{2} \)
47 \( 1 + (-4.81e5 - 8.33e5i)T + (-2.53e11 + 4.38e11i)T^{2} \)
53 \( 1 + (9.18e5 - 1.59e6i)T + (-5.87e11 - 1.01e12i)T^{2} \)
59 \( 1 + (7.25e3 - 1.25e4i)T + (-1.24e12 - 2.15e12i)T^{2} \)
61 \( 1 + (1.01e6 + 1.75e6i)T + (-1.57e12 + 2.72e12i)T^{2} \)
67 \( 1 + (1.48e6 - 2.57e6i)T + (-3.03e12 - 5.24e12i)T^{2} \)
71 \( 1 - 4.34e6T + 9.09e12T^{2} \)
73 \( 1 + (7.50e5 - 1.29e6i)T + (-5.52e12 - 9.56e12i)T^{2} \)
79 \( 1 + (8.86e5 + 1.53e6i)T + (-9.60e12 + 1.66e13i)T^{2} \)
83 \( 1 - 1.57e6T + 2.71e13T^{2} \)
89 \( 1 + (4.39e6 + 7.61e6i)T + (-2.21e13 + 3.83e13i)T^{2} \)
97 \( 1 + 1.03e7T + 8.07e13T^{2} \)
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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

−12.56195673050142795080903437922, −11.18075653294429912019775168211, −10.45341193956459400785366193956, −9.697085232870371798682006737998, −7.84426865139574111909616550133, −7.19078685226657268864656690411, −5.73515543423908211931083506663, −4.52977184293590814378155981093, −2.69816561637934675650291563587, −1.66598793371003698054693666892, 0.806425467189971276717523388570, 1.73064351829055000806950376415, 3.79938210199151210583213155027, 5.26416026578023570100250404670, 5.96236838076228912926223502571, 7.85489616837917819192432296874, 8.744177324759779041357662059554, 9.648023349185256061879782117337, 10.98083699351133545969571320900, 12.09196270190158823691422505935

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