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Properties

Label 2-45-45.14-c4-0-0
Degree $2$
Conductor $45$
Sign $0.488 - 0.872i$
Analytic cond. $4.65164$
Root an. cond. $2.15676$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.75 − 6.50i)2-s + (3.49 − 8.29i)3-s + (−20.1 + 34.9i)4-s + (2.78 + 24.8i)5-s + (−67.0 + 8.43i)6-s + (−60.4 + 34.9i)7-s + 182.·8-s + (−56.6 − 57.9i)9-s + (151. − 111. i)10-s + (−45.1 + 26.0i)11-s + (219. + 289. i)12-s + (−104. − 60.2i)13-s + (454. + 262. i)14-s + (215. + 63.6i)15-s + (−363. − 629. i)16-s − 80.3·17-s + ⋯
L(s)  = 1  + (−0.938 − 1.62i)2-s + (0.387 − 0.921i)3-s + (−1.26 + 2.18i)4-s + (0.111 + 0.993i)5-s + (−1.86 + 0.234i)6-s + (−1.23 + 0.712i)7-s + 2.85·8-s + (−0.699 − 0.715i)9-s + (1.51 − 1.11i)10-s + (−0.373 + 0.215i)11-s + (1.52 + 2.00i)12-s + (−0.617 − 0.356i)13-s + (2.31 + 1.33i)14-s + (0.959 + 0.282i)15-s + (−1.42 − 2.45i)16-s − 0.278·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.488 - 0.872i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.488 - 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $0.488 - 0.872i$
Analytic conductor: \(4.65164\)
Root analytic conductor: \(2.15676\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 45,\ (\ :2),\ 0.488 - 0.872i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.0789271 + 0.0462529i\)
\(L(\frac12)\) \(\approx\) \(0.0789271 + 0.0462529i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-3.49 + 8.29i)T \)
5 \( 1 + (-2.78 - 24.8i)T \)
good2 \( 1 + (3.75 + 6.50i)T + (-8 + 13.8i)T^{2} \)
7 \( 1 + (60.4 - 34.9i)T + (1.20e3 - 2.07e3i)T^{2} \)
11 \( 1 + (45.1 - 26.0i)T + (7.32e3 - 1.26e4i)T^{2} \)
13 \( 1 + (104. + 60.2i)T + (1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 + 80.3T + 8.35e4T^{2} \)
19 \( 1 + 254.T + 1.30e5T^{2} \)
23 \( 1 + (-125. + 216. i)T + (-1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (-250. + 144. i)T + (3.53e5 - 6.12e5i)T^{2} \)
31 \( 1 + (11.0 - 19.1i)T + (-4.61e5 - 7.99e5i)T^{2} \)
37 \( 1 - 2.47e3iT - 1.87e6T^{2} \)
41 \( 1 + (-421. - 243. i)T + (1.41e6 + 2.44e6i)T^{2} \)
43 \( 1 + (86.4 - 49.9i)T + (1.70e6 - 2.96e6i)T^{2} \)
47 \( 1 + (1.73e3 + 3.01e3i)T + (-2.43e6 + 4.22e6i)T^{2} \)
53 \( 1 - 2.91e3T + 7.89e6T^{2} \)
59 \( 1 + (1.51e3 + 875. i)T + (6.05e6 + 1.04e7i)T^{2} \)
61 \( 1 + (2.79e3 + 4.83e3i)T + (-6.92e6 + 1.19e7i)T^{2} \)
67 \( 1 + (3.32e3 + 1.92e3i)T + (1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + 988. iT - 2.54e7T^{2} \)
73 \( 1 - 6.01e3iT - 2.83e7T^{2} \)
79 \( 1 + (31.6 + 54.8i)T + (-1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 + (727. + 1.26e3i)T + (-2.37e7 + 4.11e7i)T^{2} \)
89 \( 1 - 1.28e4iT - 6.27e7T^{2} \)
97 \( 1 + (6.34e3 - 3.66e3i)T + (4.42e7 - 7.66e7i)T^{2} \)
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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

−15.13297493819187343638948613692, −13.47817980432391016986419274631, −12.65984929067754654021755079715, −11.76769913718156134827634673921, −10.39359086666434776311581184347, −9.434657314334622434432650399953, −8.164500389667454421181591880786, −6.68852942692595322596811583227, −3.16536431900863252062486750120, −2.32375443591893123504558575669, 0.07095833619824233465240157513, 4.45579540063299160254525667904, 5.86724088417133632224327945210, 7.43828658782856862930972192339, 8.790569011932571119676938050324, 9.517751637083400830602226197029, 10.46755230764206943498970972635, 13.11210624713428071469887308444, 14.14925432718035188861163771879, 15.37873472438722569604453590527

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