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Properties

Label 2-825-33.32-c1-0-15
Degree $2$
Conductor $825$
Sign $-0.877 - 0.479i$
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  − 0.715·2-s + (0.694 + 1.58i)3-s − 1.48·4-s + (−0.497 − 1.13i)6-s + 3.72i·7-s + 2.49·8-s + (−2.03 + 2.20i)9-s + (2.62 − 2.02i)11-s + (−1.03 − 2.36i)12-s + 0.930i·13-s − 2.66i·14-s + 1.19·16-s + 4.97·17-s + (1.45 − 1.57i)18-s + 6.87i·19-s + ⋯
L(s)  = 1  − 0.505·2-s + (0.401 + 0.915i)3-s − 0.744·4-s + (−0.203 − 0.463i)6-s + 1.40i·7-s + 0.882·8-s + (−0.678 + 0.735i)9-s + (0.791 − 0.611i)11-s + (−0.298 − 0.681i)12-s + 0.258i·13-s − 0.711i·14-s + 0.297·16-s + 1.20·17-s + (0.343 − 0.371i)18-s + 1.57i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.877 - 0.479i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.877 - 0.479i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $-0.877 - 0.479i$
Analytic conductor: \(6.58765\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{825} (626, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :1/2),\ -0.877 - 0.479i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.233005 + 0.912286i\)
\(L(\frac12)\) \(\approx\) \(0.233005 + 0.912286i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.694 - 1.58i)T \)
5 \( 1 \)
11 \( 1 + (-2.62 + 2.02i)T \)
good2 \( 1 + 0.715T + 2T^{2} \)
7 \( 1 - 3.72iT - 7T^{2} \)
13 \( 1 - 0.930iT - 13T^{2} \)
17 \( 1 - 4.97T + 17T^{2} \)
19 \( 1 - 6.87iT - 19T^{2} \)
23 \( 1 - 0.155iT - 23T^{2} \)
29 \( 1 + 7.90T + 29T^{2} \)
31 \( 1 + 10.0T + 31T^{2} \)
37 \( 1 - 2.19T + 37T^{2} \)
41 \( 1 - 2.75T + 41T^{2} \)
43 \( 1 + 6.10iT - 43T^{2} \)
47 \( 1 - 0.726iT - 47T^{2} \)
53 \( 1 - 8.46iT - 53T^{2} \)
59 \( 1 + 13.7iT - 59T^{2} \)
61 \( 1 - 11.5iT - 61T^{2} \)
67 \( 1 + 2.41T + 67T^{2} \)
71 \( 1 - 4.25iT - 71T^{2} \)
73 \( 1 - 5.42iT - 73T^{2} \)
79 \( 1 + 1.75iT - 79T^{2} \)
83 \( 1 - 4.89T + 83T^{2} \)
89 \( 1 + 6.91iT - 89T^{2} \)
97 \( 1 + 0.634T + 97T^{2} \)
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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

−10.30581405655094415109288834614, −9.414021041809986089007789080243, −9.120053566833484257402830757628, −8.340887901791442515327209842054, −7.59154420618616990592789522371, −5.70515295899045328411725945646, −5.52587530020052584974651871841, −4.06489537627353000647218275830, −3.38342332485439692263704271738, −1.79934873363185073784734782061, 0.57003666557250167474725016205, 1.62534564631338051313761529073, 3.40119531733813888218962899334, 4.25002428261914160121460766902, 5.46084619381066691464177211955, 6.84505604574683863154762097858, 7.42633921100332192262682524225, 7.993107668807715796660875140350, 9.214819592878177394811352914780, 9.508016069706475583960389135861

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