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Properties

Label 2-110-11.2-c2-0-7
Degree $2$
Conductor $110$
Sign $0.918 + 0.394i$
Analytic cond. $2.99728$
Root an. cond. $1.73126$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.34 + 0.437i)2-s + (4.46 − 3.24i)3-s + (1.61 + 1.17i)4-s + (−0.690 − 2.12i)5-s + (7.43 − 2.41i)6-s + (−6.89 + 9.48i)7-s + (1.66 + 2.28i)8-s + (6.65 − 20.4i)9-s − 3.16i·10-s + (−7.06 + 8.42i)11-s + 11.0·12-s + (−11.5 − 3.75i)13-s + (−13.4 + 9.75i)14-s + (−9.99 − 7.26i)15-s + (1.23 + 3.80i)16-s + (9.90 − 3.21i)17-s + ⋯
L(s)  = 1  + (0.672 + 0.218i)2-s + (1.48 − 1.08i)3-s + (0.404 + 0.293i)4-s + (−0.138 − 0.425i)5-s + (1.23 − 0.402i)6-s + (−0.984 + 1.35i)7-s + (0.207 + 0.286i)8-s + (0.739 − 2.27i)9-s − 0.316i·10-s + (−0.642 + 0.766i)11-s + 0.920·12-s + (−0.888 − 0.288i)13-s + (−0.958 + 0.696i)14-s + (−0.666 − 0.484i)15-s + (0.0772 + 0.237i)16-s + (0.582 − 0.189i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.918 + 0.394i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.918 + 0.394i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(110\)    =    \(2 \cdot 5 \cdot 11\)
Sign: $0.918 + 0.394i$
Analytic conductor: \(2.99728\)
Root analytic conductor: \(1.73126\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{110} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 110,\ (\ :1),\ 0.918 + 0.394i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.55312 - 0.525287i\)
\(L(\frac12)\) \(\approx\) \(2.55312 - 0.525287i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.34 - 0.437i)T \)
5 \( 1 + (0.690 + 2.12i)T \)
11 \( 1 + (7.06 - 8.42i)T \)
good3 \( 1 + (-4.46 + 3.24i)T + (2.78 - 8.55i)T^{2} \)
7 \( 1 + (6.89 - 9.48i)T + (-15.1 - 46.6i)T^{2} \)
13 \( 1 + (11.5 + 3.75i)T + (136. + 99.3i)T^{2} \)
17 \( 1 + (-9.90 + 3.21i)T + (233. - 169. i)T^{2} \)
19 \( 1 + (-2.98 - 4.10i)T + (-111. + 343. i)T^{2} \)
23 \( 1 - 30.9T + 529T^{2} \)
29 \( 1 + (9.06 - 12.4i)T + (-259. - 799. i)T^{2} \)
31 \( 1 + (-4.06 + 12.5i)T + (-777. - 564. i)T^{2} \)
37 \( 1 + (5.25 + 3.81i)T + (423. + 1.30e3i)T^{2} \)
41 \( 1 + (28.6 + 39.3i)T + (-519. + 1.59e3i)T^{2} \)
43 \( 1 + 35.6iT - 1.84e3T^{2} \)
47 \( 1 + (27.4 - 19.9i)T + (682. - 2.10e3i)T^{2} \)
53 \( 1 + (17.1 - 52.8i)T + (-2.27e3 - 1.65e3i)T^{2} \)
59 \( 1 + (43.5 + 31.6i)T + (1.07e3 + 3.31e3i)T^{2} \)
61 \( 1 + (-41.6 + 13.5i)T + (3.01e3 - 2.18e3i)T^{2} \)
67 \( 1 - 117.T + 4.48e3T^{2} \)
71 \( 1 + (-28.7 - 88.3i)T + (-4.07e3 + 2.96e3i)T^{2} \)
73 \( 1 + (0.746 - 1.02i)T + (-1.64e3 - 5.06e3i)T^{2} \)
79 \( 1 + (106. + 34.7i)T + (5.04e3 + 3.66e3i)T^{2} \)
83 \( 1 + (-123. + 39.9i)T + (5.57e3 - 4.04e3i)T^{2} \)
89 \( 1 + 4.92T + 7.92e3T^{2} \)
97 \( 1 + (5.54 - 17.0i)T + (-7.61e3 - 5.53e3i)T^{2} \)
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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

−13.13371898327993070254622445806, −12.63256172080841627895505184274, −12.08819816764119358079691007664, −9.727914435807704373946261333573, −8.857081022304752244371186627722, −7.74268122690572419030733411390, −6.83704694284183104522080197648, −5.32653819713127026642450933780, −3.24760188555443909633835722780, −2.29647232057732964397598201692, 2.92010160426835003862580117246, 3.59696306623788698002842223491, 4.85930817308458427494575243933, 6.93090016531228127838911712223, 8.010648588541247434400350435461, 9.557246882086547755342871288120, 10.20130788457167196801272776116, 11.07653212485481610764799374318, 12.99241411902804982073385472630, 13.63572865134580812005484493974

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