Properties

Label 2-112-7.3-c4-0-2
Degree $2$
Conductor $112$
Sign $0.961 + 0.275i$
Analytic cond. $11.5774$
Root an. cond. $3.40256$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−13.6 − 7.89i)3-s + (−24.9 + 14.3i)5-s + (−39.2 + 29.2i)7-s + (84.0 + 145. i)9-s + (−6.39 + 11.0i)11-s − 283. i·13-s + 453.·15-s + (−164. − 94.7i)17-s + (139. − 80.7i)19-s + (768. − 90.5i)21-s + (86.2 + 149. i)23-s + (101. − 175. i)25-s − 1.37e3i·27-s + 711.·29-s + (747. + 431. i)31-s + ⋯
L(s)  = 1  + (−1.51 − 0.876i)3-s + (−0.996 + 0.575i)5-s + (−0.801 + 0.597i)7-s + (1.03 + 1.79i)9-s + (−0.0528 + 0.0915i)11-s − 1.67i·13-s + 2.01·15-s + (−0.567 − 0.327i)17-s + (0.387 − 0.223i)19-s + (1.74 − 0.205i)21-s + (0.163 + 0.282i)23-s + (0.161 − 0.280i)25-s − 1.88i·27-s + 0.845·29-s + (0.777 + 0.449i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.961 + 0.275i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.961 + 0.275i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $0.961 + 0.275i$
Analytic conductor: \(11.5774\)
Root analytic conductor: \(3.40256\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{112} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 112,\ (\ :2),\ 0.961 + 0.275i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.531388 - 0.0747524i\)
\(L(\frac12)\) \(\approx\) \(0.531388 - 0.0747524i\)
\(L(3)\) 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 \( 1 + (39.2 - 29.2i)T \)
good3 \( 1 + (13.6 + 7.89i)T + (40.5 + 70.1i)T^{2} \)
5 \( 1 + (24.9 - 14.3i)T + (312.5 - 541. i)T^{2} \)
11 \( 1 + (6.39 - 11.0i)T + (-7.32e3 - 1.26e4i)T^{2} \)
13 \( 1 + 283. iT - 2.85e4T^{2} \)
17 \( 1 + (164. + 94.7i)T + (4.17e4 + 7.23e4i)T^{2} \)
19 \( 1 + (-139. + 80.7i)T + (6.51e4 - 1.12e5i)T^{2} \)
23 \( 1 + (-86.2 - 149. i)T + (-1.39e5 + 2.42e5i)T^{2} \)
29 \( 1 - 711.T + 7.07e5T^{2} \)
31 \( 1 + (-747. - 431. i)T + (4.61e5 + 7.99e5i)T^{2} \)
37 \( 1 + (-411. - 712. i)T + (-9.37e5 + 1.62e6i)T^{2} \)
41 \( 1 - 2.85e3iT - 2.82e6T^{2} \)
43 \( 1 + 1.32e3T + 3.41e6T^{2} \)
47 \( 1 + (-2.99e3 + 1.73e3i)T + (2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + (102. - 177. i)T + (-3.94e6 - 6.83e6i)T^{2} \)
59 \( 1 + (3.35e3 + 1.93e3i)T + (6.05e6 + 1.04e7i)T^{2} \)
61 \( 1 + (-1.78e3 + 1.03e3i)T + (6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (-63.1 + 109. i)T + (-1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 - 7.59e3T + 2.54e7T^{2} \)
73 \( 1 + (4.44e3 + 2.56e3i)T + (1.41e7 + 2.45e7i)T^{2} \)
79 \( 1 + (1.35e3 + 2.35e3i)T + (-1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 - 7.84e3iT - 4.74e7T^{2} \)
89 \( 1 + (5.59e3 - 3.23e3i)T + (3.13e7 - 5.43e7i)T^{2} \)
97 \( 1 + 189. iT - 8.85e7T^{2} \)
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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

−12.60520176535789067692567458072, −11.85773927345599497657841910429, −11.06765414857631435209908077256, −10.05481647665336246401250276651, −8.123681656537770568785653374544, −7.08971117732502057168053301925, −6.19301385072252099152011302981, −5.03999948136930433612131174902, −3.00205540084915751160584861583, −0.61362940374380709804810233126, 0.57594052357360818505390051962, 3.98225583784463100409438291909, 4.55269387988502647471851602085, 6.10136391847877235752585141272, 7.10866611052735037663320462377, 8.892376342693149674658127002880, 9.979149657098770174542196802092, 10.98396021110782420354014452644, 11.82730611525970109645358198756, 12.50004690986403807210644781439

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