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Properties

Label 2-45-15.2-c3-0-5
Degree $2$
Conductor $45$
Sign $-0.0106 + 0.999i$
Analytic cond. $2.65508$
Root an. cond. $1.62944$
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  + (3.74 − 3.74i)2-s − 20.0i·4-s + (0.572 + 11.1i)5-s + (1.80 + 1.80i)7-s + (−45.3 − 45.3i)8-s + (43.9 + 39.7i)10-s + 46.0i·11-s + (18.6 − 18.6i)13-s + 13.5·14-s − 178.·16-s + (−14.5 + 14.5i)17-s − 74.4i·19-s + (224. − 11.4i)20-s + (172. + 172. i)22-s + (59.6 + 59.6i)23-s + ⋯
L(s)  = 1  + (1.32 − 1.32i)2-s − 2.51i·4-s + (0.0511 + 0.998i)5-s + (0.0977 + 0.0977i)7-s + (−2.00 − 2.00i)8-s + (1.39 + 1.25i)10-s + 1.26i·11-s + (0.397 − 0.397i)13-s + 0.258·14-s − 2.79·16-s + (−0.208 + 0.208i)17-s − 0.899i·19-s + (2.50 − 0.128i)20-s + (1.67 + 1.67i)22-s + (0.540 + 0.540i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0106 + 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0106 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $-0.0106 + 0.999i$
Analytic conductor: \(2.65508\)
Root analytic conductor: \(1.62944\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 45,\ (\ :3/2),\ -0.0106 + 0.999i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.69188 - 1.71006i\)
\(L(\frac12)\) \(\approx\) \(1.69188 - 1.71006i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (-0.572 - 11.1i)T \)
good2 \( 1 + (-3.74 + 3.74i)T - 8iT^{2} \)
7 \( 1 + (-1.80 - 1.80i)T + 343iT^{2} \)
11 \( 1 - 46.0iT - 1.33e3T^{2} \)
13 \( 1 + (-18.6 + 18.6i)T - 2.19e3iT^{2} \)
17 \( 1 + (14.5 - 14.5i)T - 4.91e3iT^{2} \)
19 \( 1 + 74.4iT - 6.85e3T^{2} \)
23 \( 1 + (-59.6 - 59.6i)T + 1.21e4iT^{2} \)
29 \( 1 + 202.T + 2.43e4T^{2} \)
31 \( 1 + 49.5T + 2.97e4T^{2} \)
37 \( 1 + (45.0 + 45.0i)T + 5.06e4iT^{2} \)
41 \( 1 + 306. iT - 6.89e4T^{2} \)
43 \( 1 + (230. - 230. i)T - 7.95e4iT^{2} \)
47 \( 1 + (-176. + 176. i)T - 1.03e5iT^{2} \)
53 \( 1 + (-85.1 - 85.1i)T + 1.48e5iT^{2} \)
59 \( 1 + 330.T + 2.05e5T^{2} \)
61 \( 1 - 678.T + 2.26e5T^{2} \)
67 \( 1 + (-756. - 756. i)T + 3.00e5iT^{2} \)
71 \( 1 + 100. iT - 3.57e5T^{2} \)
73 \( 1 + (-586. + 586. i)T - 3.89e5iT^{2} \)
79 \( 1 - 286. iT - 4.93e5T^{2} \)
83 \( 1 + (-947. - 947. i)T + 5.71e5iT^{2} \)
89 \( 1 + 688.T + 7.04e5T^{2} \)
97 \( 1 + (-920. - 920. i)T + 9.12e5iT^{2} \)
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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

−14.88379317799305946326652501700, −13.65875005661878152586187336628, −12.75067901659105520433310074236, −11.52155327879237657645196057469, −10.69238332041860474635018204742, −9.596595377498131662661224838214, −6.92915241898152375360314655596, −5.33611006690288300887331014665, −3.73065969240681687381356584178, −2.19507363117433807024548329293, 3.76662165156776215831880057157, 5.18020765190658793377745707212, 6.28136063266477399292457081490, 7.901567943343437910334278050316, 8.901861532890001348050120309247, 11.43352126077563228260501642352, 12.66884719790986681496756750712, 13.51416083185151146875026278915, 14.38713671560380183243925954209, 15.67292759047338541742925285243

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