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Properties

Label 2-112-112.109-c1-0-4
Degree $2$
Conductor $112$
Sign $0.761 - 0.648i$
Analytic cond. $0.894324$
Root an. cond. $0.945687$
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  + (−0.805 + 1.16i)2-s + (1.16 − 0.312i)3-s + (−0.703 − 1.87i)4-s + (0.980 + 0.262i)5-s + (−0.575 + 1.60i)6-s + (2.60 − 0.476i)7-s + (2.74 + 0.690i)8-s + (−1.33 + 0.772i)9-s + (−1.09 + 0.928i)10-s + (0.635 + 2.36i)11-s + (−1.40 − 1.96i)12-s + (−2.65 − 2.65i)13-s + (−1.54 + 3.40i)14-s + 1.22·15-s + (−3.01 + 2.63i)16-s + (−0.509 + 0.881i)17-s + ⋯
L(s)  = 1  + (−0.569 + 0.822i)2-s + (0.672 − 0.180i)3-s + (−0.351 − 0.936i)4-s + (0.438 + 0.117i)5-s + (−0.234 + 0.655i)6-s + (0.983 − 0.180i)7-s + (0.969 + 0.243i)8-s + (−0.445 + 0.257i)9-s + (−0.346 + 0.293i)10-s + (0.191 + 0.714i)11-s + (−0.405 − 0.566i)12-s + (−0.737 − 0.737i)13-s + (−0.412 + 0.911i)14-s + 0.316·15-s + (−0.752 + 0.658i)16-s + (−0.123 + 0.213i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.761 - 0.648i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.761 - 0.648i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $0.761 - 0.648i$
Analytic conductor: \(0.894324\)
Root analytic conductor: \(0.945687\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{112} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 112,\ (\ :1/2),\ 0.761 - 0.648i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.951721 + 0.350294i\)
\(L(\frac12)\) \(\approx\) \(0.951721 + 0.350294i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.805 - 1.16i)T \)
7 \( 1 + (-2.60 + 0.476i)T \)
good3 \( 1 + (-1.16 + 0.312i)T + (2.59 - 1.5i)T^{2} \)
5 \( 1 + (-0.980 - 0.262i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-0.635 - 2.36i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (2.65 + 2.65i)T + 13iT^{2} \)
17 \( 1 + (0.509 - 0.881i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.0250 - 0.0936i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-1.67 + 0.965i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (5.05 + 5.05i)T + 29iT^{2} \)
31 \( 1 + (-4.28 + 7.41i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (7.71 + 2.06i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 8.51iT - 41T^{2} \)
43 \( 1 + (4.47 - 4.47i)T - 43iT^{2} \)
47 \( 1 + (-6.02 - 10.4i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.381 - 1.42i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (1.86 + 6.96i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (1.18 - 4.42i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (3.38 - 0.907i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 5.43iT - 71T^{2} \)
73 \( 1 + (-7.34 - 4.23i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.433 + 0.751i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.44 + 5.44i)T + 83iT^{2} \)
89 \( 1 + (-3.93 + 2.26i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 16.4T + 97T^{2} \)
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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

−14.07352283390473471330371790820, −13.12979934051783227218153846251, −11.46591577927559102451795722631, −10.26708978753142569592869788352, −9.310278999848501196552339297519, −8.081526228461949060354867491066, −7.52888208813809540439983097762, −5.94896616950191879520144207943, −4.67496610897568866798771487703, −2.11742024501098861949910709069, 2.00037920703147453178741627006, 3.53620653991887187912954705945, 5.17809230821962369654177401241, 7.24221832663171625247712787018, 8.650157249104084124469311445452, 9.013336589524740243942392935390, 10.29774146121216466199636604962, 11.44960390368580032184553226778, 12.16389784702856730390662413623, 13.71573686372533739646868299230

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