Properties

Label 2-112-112.109-c1-0-9
Degree $2$
Conductor $112$
Sign $0.365 + 0.930i$
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  + (0.236 − 1.39i)2-s + (0.827 − 0.221i)3-s + (−1.88 − 0.659i)4-s + (4.10 + 1.10i)5-s + (−0.113 − 1.20i)6-s + (−2.50 − 0.856i)7-s + (−1.36 + 2.47i)8-s + (−1.96 + 1.13i)9-s + (2.50 − 5.46i)10-s + (−0.318 − 1.18i)11-s + (−1.70 − 0.127i)12-s + (−1.73 − 1.73i)13-s + (−1.78 + 3.28i)14-s + 3.64·15-s + (3.12 + 2.49i)16-s + (−0.931 + 1.61i)17-s + ⋯
L(s)  = 1  + (0.167 − 0.985i)2-s + (0.477 − 0.128i)3-s + (−0.944 − 0.329i)4-s + (1.83 + 0.492i)5-s + (−0.0463 − 0.492i)6-s + (−0.946 − 0.323i)7-s + (−0.483 + 0.875i)8-s + (−0.653 + 0.377i)9-s + (0.792 − 1.72i)10-s + (−0.0960 − 0.358i)11-s + (−0.493 − 0.0367i)12-s + (−0.480 − 0.480i)13-s + (−0.477 + 0.878i)14-s + 0.941·15-s + (0.782 + 0.622i)16-s + (−0.226 + 0.391i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.365 + 0.930i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.365 + 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $0.365 + 0.930i$
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: \((2,\ 112,\ (\ :1/2),\ 0.365 + 0.930i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.07234 - 0.730674i\)
\(L(\frac12)\) \(\approx\) \(1.07234 - 0.730674i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.236 + 1.39i)T \)
7 \( 1 + (2.50 + 0.856i)T \)
good3 \( 1 + (-0.827 + 0.221i)T + (2.59 - 1.5i)T^{2} \)
5 \( 1 + (-4.10 - 1.10i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (0.318 + 1.18i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (1.73 + 1.73i)T + 13iT^{2} \)
17 \( 1 + (0.931 - 1.61i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.989 - 3.69i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-1.23 + 0.711i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.181 + 0.181i)T + 29iT^{2} \)
31 \( 1 + (-3.23 + 5.59i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.237 - 0.0637i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 0.440iT - 41T^{2} \)
43 \( 1 + (5.54 - 5.54i)T - 43iT^{2} \)
47 \( 1 + (3.61 + 6.26i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.59 + 9.69i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (0.438 + 1.63i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-2.41 + 9.01i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (-9.59 + 2.57i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 11.2iT - 71T^{2} \)
73 \( 1 + (8.13 + 4.69i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.52 - 11.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.06 + 3.06i)T + 83iT^{2} \)
89 \( 1 + (5.66 - 3.26i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 1.70T + 97T^{2} \)
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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

−13.33604085354823084713835919561, −12.80778720741304024736171073852, −11.12778441881589038072181549834, −10.10963551675717091057975209578, −9.626913169972169685853831932776, −8.365371306920116968752507478607, −6.37672256914359880669981747264, −5.39909432672569651078986897397, −3.23318629645424331622198228323, −2.18620792825539896361286342099, 2.77951624861275821114461978880, 4.89599327448056712180847794046, 5.99767104653003308159553400900, 6.86209508217815975867824533758, 8.790346963326190625961927317539, 9.258470421225472381156409325817, 10.05342084323695897180433000781, 12.25635735068439559345704059712, 13.22389477910623338986127915785, 13.83881648241639632068327605590

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