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Properties

Label 2-405-45.34-c1-0-12
Degree $2$
Conductor $405$
Sign $-0.476 + 0.879i$
Analytic cond. $3.23394$
Root an. cond. $1.79831$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.22 − 0.707i)2-s + (−0.448 − 2.19i)5-s + (2.59 + 1.5i)7-s + 2.82i·8-s + (−1 + 2.99i)10-s + (2.12 − 3.67i)11-s + (2.59 − 1.5i)13-s + (−2.12 − 3.67i)14-s + (2.00 − 3.46i)16-s − 2.82i·17-s − 19-s + (−5.19 + 3i)22-s + (−6.12 + 3.53i)23-s + (−4.59 + 1.96i)25-s − 4.24·26-s + ⋯
L(s)  = 1  + (−0.866 − 0.499i)2-s + (−0.200 − 0.979i)5-s + (0.981 + 0.566i)7-s + 0.999i·8-s + (−0.316 + 0.948i)10-s + (0.639 − 1.10i)11-s + (0.720 − 0.416i)13-s + (−0.566 − 0.981i)14-s + (0.500 − 0.866i)16-s − 0.685i·17-s − 0.229·19-s + (−1.10 + 0.639i)22-s + (−1.27 + 0.737i)23-s + (−0.919 + 0.392i)25-s − 0.832·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.476 + 0.879i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.476 + 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $-0.476 + 0.879i$
Analytic conductor: \(3.23394\)
Root analytic conductor: \(1.79831\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :1/2),\ -0.476 + 0.879i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.430296 - 0.722329i\)
\(L(\frac12)\) \(\approx\) \(0.430296 - 0.722329i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.448 + 2.19i)T \)
good2 \( 1 + (1.22 + 0.707i)T + (1 + 1.73i)T^{2} \)
7 \( 1 + (-2.59 - 1.5i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.12 + 3.67i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.59 + 1.5i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 2.82iT - 17T^{2} \)
19 \( 1 + T + 19T^{2} \)
23 \( 1 + (6.12 - 3.53i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.12 + 3.67i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1 + 1.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 9iT - 37T^{2} \)
41 \( 1 + (2.12 + 3.67i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.19 + 3i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.44 - 1.41i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 9.89iT - 53T^{2} \)
59 \( 1 + (-4.24 - 7.34i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.5 + 11.2i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.59 + 1.5i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 12.7T + 71T^{2} \)
73 \( 1 - 9iT - 73T^{2} \)
79 \( 1 + (2.5 - 4.33i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.22 + 0.707i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + (-2.59 - 1.5i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.13909164091295481024589458278, −9.914641389458017228359356276668, −8.984296637690214887112137100129, −8.482691400341184848801660615222, −7.80453292734435524920609826251, −5.88914286965259600759547521591, −5.25277899724708434073001718399, −3.90402059407269633201049804318, −2.04446764909690950970508946416, −0.801129662420497264970198870229, 1.68823999176220550534605275916, 3.67148859113993749577723387597, 4.51390673864806631627567795241, 6.44861066936021542415933671987, 6.92426673840184250560182982203, 7.991625300352476129432568636045, 8.500906199013635890535999946627, 9.832849069882315832683557039177, 10.38833015051355673899477000273, 11.38775438797652661230251385468

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