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Properties

Label 2-405-45.14-c2-0-1
Degree $2$
Conductor $405$
Sign $0.381 - 0.924i$
Analytic cond. $11.0354$
Root an. cond. $3.32196$
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  + (−1.14 − 1.97i)2-s + (−0.608 + 1.05i)4-s + (−4.77 − 1.49i)5-s + (2.98 − 1.72i)7-s − 6.35·8-s + (2.48 + 11.1i)10-s + (3.89 − 2.25i)11-s + (−10.2 − 5.92i)13-s + (−6.81 − 3.93i)14-s + (9.69 + 16.7i)16-s − 23.3·17-s + 11.0·19-s + (4.47 − 4.11i)20-s + (−8.90 − 5.14i)22-s + (−14.9 + 25.8i)23-s + ⋯
L(s)  = 1  + (−0.570 − 0.988i)2-s + (−0.152 + 0.263i)4-s + (−0.954 − 0.299i)5-s + (0.426 − 0.246i)7-s − 0.794·8-s + (0.248 + 1.11i)10-s + (0.354 − 0.204i)11-s + (−0.788 − 0.455i)13-s + (−0.486 − 0.281i)14-s + (0.605 + 1.04i)16-s − 1.37·17-s + 0.582·19-s + (0.223 − 0.205i)20-s + (−0.404 − 0.233i)22-s + (−0.648 + 1.12i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.381 - 0.924i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.381 - 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $0.381 - 0.924i$
Analytic conductor: \(11.0354\)
Root analytic conductor: \(3.32196\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (134, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :1),\ 0.381 - 0.924i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0714610 + 0.0478108i\)
\(L(\frac12)\) \(\approx\) \(0.0714610 + 0.0478108i\)
\(L(2)\) 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 \( 1 + (4.77 + 1.49i)T \)
good2 \( 1 + (1.14 + 1.97i)T + (-2 + 3.46i)T^{2} \)
7 \( 1 + (-2.98 + 1.72i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (-3.89 + 2.25i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (10.2 + 5.92i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + 23.3T + 289T^{2} \)
19 \( 1 - 11.0T + 361T^{2} \)
23 \( 1 + (14.9 - 25.8i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (-30.8 + 17.8i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (15.1 - 26.2i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 - 5.11iT - 1.36e3T^{2} \)
41 \( 1 + (19.6 + 11.3i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (58.8 - 33.9i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-23.1 - 40.0i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 68.0T + 2.80e3T^{2} \)
59 \( 1 + (29.9 + 17.2i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (8.68 + 15.0i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-85.1 - 49.1i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 134. iT - 5.04e3T^{2} \)
73 \( 1 + 134. iT - 5.32e3T^{2} \)
79 \( 1 + (24.6 + 42.7i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (14.3 + 24.8i)T + (-3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 65.9iT - 7.92e3T^{2} \)
97 \( 1 + (62.1 - 35.8i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−11.30632322218520149490497588581, −10.41870231125074457254112617172, −9.504057816008274722400336728744, −8.619704230590767074613099424890, −7.78440446101532224765179212357, −6.65849359857797805310471478422, −5.20100429367359363098711884145, −4.03547342930591460321018074243, −2.85076629806619058011807535118, −1.35789314714633981520364170925, 0.04656201867969718252301469234, 2.48115174073555108943487416584, 3.99895278365768910756670153714, 5.13380433924773323791370790999, 6.66197703356373634339182502110, 7.01336380969877424288455926039, 8.148452020556462933296770618540, 8.659216565763136889635798859898, 9.697154916699353963561931793666, 10.89118117340673126823878415182

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