Properties

Label 2-110-11.5-c3-0-4
Degree $2$
Conductor $110$
Sign $0.915 - 0.401i$
Analytic cond. $6.49021$
Root an. cond. $2.54758$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.618 − 1.90i)2-s + (5.32 + 3.86i)3-s + (−3.23 + 2.35i)4-s + (−1.54 + 4.75i)5-s + (4.06 − 12.5i)6-s + (7.43 − 5.40i)7-s + (6.47 + 4.70i)8-s + (5.04 + 15.5i)9-s + 10.0·10-s + (32.2 + 16.9i)11-s − 26.3·12-s + (24.6 + 75.7i)13-s + (−14.8 − 10.8i)14-s + (−26.6 + 19.3i)15-s + (4.94 − 15.2i)16-s + (−5.70 + 17.5i)17-s + ⋯
L(s)  = 1  + (−0.218 − 0.672i)2-s + (1.02 + 0.744i)3-s + (−0.404 + 0.293i)4-s + (−0.138 + 0.425i)5-s + (0.276 − 0.851i)6-s + (0.401 − 0.291i)7-s + (0.286 + 0.207i)8-s + (0.186 + 0.575i)9-s + 0.316·10-s + (0.885 + 0.465i)11-s − 0.633·12-s + (0.525 + 1.61i)13-s + (−0.284 − 0.206i)14-s + (−0.458 + 0.333i)15-s + (0.0772 − 0.237i)16-s + (−0.0813 + 0.250i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.915 - 0.401i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.915 - 0.401i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(110\)    =    \(2 \cdot 5 \cdot 11\)
Sign: $0.915 - 0.401i$
Analytic conductor: \(6.49021\)
Root analytic conductor: \(2.54758\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{110} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 110,\ (\ :3/2),\ 0.915 - 0.401i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.90944 + 0.399816i\)
\(L(\frac12)\) \(\approx\) \(1.90944 + 0.399816i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.618 + 1.90i)T \)
5 \( 1 + (1.54 - 4.75i)T \)
11 \( 1 + (-32.2 - 16.9i)T \)
good3 \( 1 + (-5.32 - 3.86i)T + (8.34 + 25.6i)T^{2} \)
7 \( 1 + (-7.43 + 5.40i)T + (105. - 326. i)T^{2} \)
13 \( 1 + (-24.6 - 75.7i)T + (-1.77e3 + 1.29e3i)T^{2} \)
17 \( 1 + (5.70 - 17.5i)T + (-3.97e3 - 2.88e3i)T^{2} \)
19 \( 1 + (-12.9 - 9.41i)T + (2.11e3 + 6.52e3i)T^{2} \)
23 \( 1 - 7.44T + 1.21e4T^{2} \)
29 \( 1 + (-44.2 + 32.1i)T + (7.53e3 - 2.31e4i)T^{2} \)
31 \( 1 + (55.5 + 170. i)T + (-2.41e4 + 1.75e4i)T^{2} \)
37 \( 1 + (-133. + 96.7i)T + (1.56e4 - 4.81e4i)T^{2} \)
41 \( 1 + (179. + 130. i)T + (2.12e4 + 6.55e4i)T^{2} \)
43 \( 1 + 115.T + 7.95e4T^{2} \)
47 \( 1 + (380. + 276. i)T + (3.20e4 + 9.87e4i)T^{2} \)
53 \( 1 + (224. + 691. i)T + (-1.20e5 + 8.75e4i)T^{2} \)
59 \( 1 + (114. - 83.1i)T + (6.34e4 - 1.95e5i)T^{2} \)
61 \( 1 + (191. - 590. i)T + (-1.83e5 - 1.33e5i)T^{2} \)
67 \( 1 - 783.T + 3.00e5T^{2} \)
71 \( 1 + (99.8 - 307. i)T + (-2.89e5 - 2.10e5i)T^{2} \)
73 \( 1 + (-806. + 586. i)T + (1.20e5 - 3.69e5i)T^{2} \)
79 \( 1 + (-54.1 - 166. i)T + (-3.98e5 + 2.89e5i)T^{2} \)
83 \( 1 + (-214. + 659. i)T + (-4.62e5 - 3.36e5i)T^{2} \)
89 \( 1 + 101.T + 7.04e5T^{2} \)
97 \( 1 + (-16.1 - 49.6i)T + (-7.38e5 + 5.36e5i)T^{2} \)
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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

−13.47881450648250839791448385984, −11.91374802976151544669028208453, −11.13323259048964494915362248490, −9.863337218339843916041457557804, −9.178836339133585218344912050477, −8.167067312005438211390957464408, −6.70706727599015435209827724511, −4.38498764097170279568610455069, −3.61806282192386613066974159009, −1.93252453882285229670585470730, 1.23185427506815647531564093111, 3.23650743115899426341393550059, 5.15462465326309848761268680779, 6.57012918759035079529729605687, 7.946636753421089880194442560691, 8.392619254499830148854726436639, 9.416407151085666820401739923210, 10.99196196346646940016724184708, 12.42705799078598520418494248061, 13.33156520281787727653218881462

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