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Properties

Label 2-825-165.38-c1-0-60
Degree $2$
Conductor $825$
Sign $-0.997 - 0.0691i$
Analytic cond. $6.58765$
Root an. cond. $2.56664$
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  + (0.754 − 1.48i)2-s + (−0.647 − 1.60i)3-s + (−0.448 − 0.617i)4-s + (−2.86 − 0.253i)6-s + (0.185 − 1.17i)7-s + (2.03 − 0.321i)8-s + (−2.16 + 2.08i)9-s + (−1.17 − 3.10i)11-s + (−0.700 + 1.11i)12-s + (2.25 − 4.41i)13-s + (−1.59 − 1.16i)14-s + (1.52 − 4.70i)16-s + (−0.807 + 0.411i)17-s + (1.44 + 4.77i)18-s + (−1.28 + 1.76i)19-s + ⋯
L(s)  = 1  + (0.533 − 1.04i)2-s + (−0.373 − 0.927i)3-s + (−0.224 − 0.308i)4-s + (−1.17 − 0.103i)6-s + (0.0702 − 0.443i)7-s + (0.718 − 0.113i)8-s + (−0.720 + 0.693i)9-s + (−0.352 − 0.935i)11-s + (−0.202 + 0.323i)12-s + (0.624 − 1.22i)13-s + (−0.426 − 0.310i)14-s + (0.381 − 1.17i)16-s + (−0.195 + 0.0997i)17-s + (0.341 + 1.12i)18-s + (−0.294 + 0.405i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0691i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0691i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $-0.997 - 0.0691i$
Analytic conductor: \(6.58765\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{825} (368, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :1/2),\ -0.997 - 0.0691i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0594672 + 1.71909i\)
\(L(\frac12)\) \(\approx\) \(0.0594672 + 1.71909i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.647 + 1.60i)T \)
5 \( 1 \)
11 \( 1 + (1.17 + 3.10i)T \)
good2 \( 1 + (-0.754 + 1.48i)T + (-1.17 - 1.61i)T^{2} \)
7 \( 1 + (-0.185 + 1.17i)T + (-6.65 - 2.16i)T^{2} \)
13 \( 1 + (-2.25 + 4.41i)T + (-7.64 - 10.5i)T^{2} \)
17 \( 1 + (0.807 - 0.411i)T + (9.99 - 13.7i)T^{2} \)
19 \( 1 + (1.28 - 1.76i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 + (2.79 - 2.79i)T - 23iT^{2} \)
29 \( 1 + (4.83 - 3.51i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (1.83 + 5.65i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-1.26 + 7.99i)T + (-35.1 - 11.4i)T^{2} \)
41 \( 1 + (-0.383 + 0.527i)T + (-12.6 - 38.9i)T^{2} \)
43 \( 1 + (3.51 + 3.51i)T + 43iT^{2} \)
47 \( 1 + (-0.779 - 4.92i)T + (-44.6 + 14.5i)T^{2} \)
53 \( 1 + (-10.2 - 5.20i)T + (31.1 + 42.8i)T^{2} \)
59 \( 1 + (11.4 - 8.32i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (1.08 - 3.34i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-2.16 + 2.16i)T - 67iT^{2} \)
71 \( 1 + (-7.10 - 2.30i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (-0.419 - 0.0664i)T + (69.4 + 22.5i)T^{2} \)
79 \( 1 + (-12.6 + 4.10i)T + (63.9 - 46.4i)T^{2} \)
83 \( 1 + (6.49 + 12.7i)T + (-48.7 + 67.1i)T^{2} \)
89 \( 1 - 10.5T + 89T^{2} \)
97 \( 1 + (-9.96 - 5.07i)T + (57.0 + 78.4i)T^{2} \)
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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

−10.40418603906400792507063483888, −8.921925377666621531371675954077, −7.82508171164510183666153825314, −7.44477763283089846756005370419, −6.01274036181185309175095653237, −5.44700338261789424496024093641, −4.01800053908056729303369175250, −3.13219443309236874788939644374, −1.99785525177708654056193401388, −0.74628468339780676593805779226, 2.07852068566194199322112789165, 3.82070541202570026999328495634, 4.65528412386451666686954529309, 5.27492557450969933046225753011, 6.33653296918764172229093695511, 6.81764177489242842955869013842, 8.055851921912167229755668664917, 8.922177074204627940786947560817, 9.788304605110558520828930382381, 10.62408110679527560299282552862

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