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Properties

Label 2-825-165.38-c1-0-33
Degree $2$
Conductor $825$
Sign $0.788 - 0.614i$
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  + (−1.12 + 2.20i)2-s + (−0.382 + 1.68i)3-s + (−2.43 − 3.35i)4-s + (−3.30 − 2.74i)6-s + (−0.134 + 0.847i)7-s + (5.26 − 0.833i)8-s + (−2.70 − 1.29i)9-s + (−3.29 + 0.418i)11-s + (6.60 − 2.83i)12-s + (2.14 − 4.20i)13-s + (−1.72 − 1.25i)14-s + (−1.52 + 4.67i)16-s + (−4.82 + 2.45i)17-s + (5.90 − 4.52i)18-s + (2.92 − 4.03i)19-s + ⋯
L(s)  = 1  + (−0.796 + 1.56i)2-s + (−0.220 + 0.975i)3-s + (−1.21 − 1.67i)4-s + (−1.34 − 1.12i)6-s + (−0.0507 + 0.320i)7-s + (1.86 − 0.294i)8-s + (−0.902 − 0.430i)9-s + (−0.992 + 0.126i)11-s + (1.90 − 0.818i)12-s + (0.594 − 1.16i)13-s + (−0.459 − 0.334i)14-s + (−0.380 + 1.16i)16-s + (−1.17 + 0.596i)17-s + (1.39 − 1.06i)18-s + (0.672 − 0.924i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.788 - 0.614i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.788 - 0.614i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $0.788 - 0.614i$
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: \((2,\ 825,\ (\ :1/2),\ 0.788 - 0.614i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.402631 + 0.138318i\)
\(L(\frac12)\) \(\approx\) \(0.402631 + 0.138318i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.382 - 1.68i)T \)
5 \( 1 \)
11 \( 1 + (3.29 - 0.418i)T \)
good2 \( 1 + (1.12 - 2.20i)T + (-1.17 - 1.61i)T^{2} \)
7 \( 1 + (0.134 - 0.847i)T + (-6.65 - 2.16i)T^{2} \)
13 \( 1 + (-2.14 + 4.20i)T + (-7.64 - 10.5i)T^{2} \)
17 \( 1 + (4.82 - 2.45i)T + (9.99 - 13.7i)T^{2} \)
19 \( 1 + (-2.92 + 4.03i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 + (-1.93 + 1.93i)T - 23iT^{2} \)
29 \( 1 + (-0.756 + 0.549i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (1.95 + 6.02i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (1.46 - 9.22i)T + (-35.1 - 11.4i)T^{2} \)
41 \( 1 + (-2.90 + 4.00i)T + (-12.6 - 38.9i)T^{2} \)
43 \( 1 + (-2.32 - 2.32i)T + 43iT^{2} \)
47 \( 1 + (1.85 + 11.6i)T + (-44.6 + 14.5i)T^{2} \)
53 \( 1 + (-1.46 - 0.748i)T + (31.1 + 42.8i)T^{2} \)
59 \( 1 + (2.46 - 1.79i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-3.12 + 9.60i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (6.03 - 6.03i)T - 67iT^{2} \)
71 \( 1 + (1.61 + 0.524i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (11.1 + 1.77i)T + (69.4 + 22.5i)T^{2} \)
79 \( 1 + (11.4 - 3.72i)T + (63.9 - 46.4i)T^{2} \)
83 \( 1 + (6.17 + 12.1i)T + (-48.7 + 67.1i)T^{2} \)
89 \( 1 - 14.4T + 89T^{2} \)
97 \( 1 + (-8.91 - 4.54i)T + (57.0 + 78.4i)T^{2} \)
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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

−10.18150636401278174460838504838, −9.183894335954055268807829732427, −8.614456696418918959067049127601, −7.910320549751350566971602066432, −6.86283620874007390996877374686, −5.89562407562137183468172657271, −5.33373184077148490249073393148, −4.44052754514695351852951921293, −2.88253267745271824732647051231, −0.31883706020641591398209665487, 1.17780079737868352213254220042, 2.19236583966338087804604984058, 3.17712234042684700656461285824, 4.40652055402362939863389604893, 5.75102820088938495168067232034, 7.02505043813662595889251529919, 7.74076419996198184610344487392, 8.743495101658268618006318431307, 9.248392767284018760509510782290, 10.42500463947096733807805713307

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