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Properties

Label 2-525-105.23-c1-0-27
Degree $2$
Conductor $525$
Sign $0.702 + 0.711i$
Analytic cond. $4.19214$
Root an. cond. $2.04747$
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  + (−0.298 + 0.0799i)2-s + (1.64 − 0.540i)3-s + (−1.64 + 0.952i)4-s + (−0.447 + 0.292i)6-s + (−0.951 − 2.46i)7-s + (0.852 − 0.852i)8-s + (2.41 − 1.77i)9-s + (0.660 − 0.381i)11-s + (−2.19 + 2.45i)12-s + (2.27 + 2.27i)13-s + (0.481 + 0.660i)14-s + (1.71 − 2.97i)16-s + (1.25 − 4.69i)17-s + (−0.578 + 0.723i)18-s + (1.41 + 0.818i)19-s + ⋯
L(s)  = 1  + (−0.210 + 0.0565i)2-s + (0.950 − 0.312i)3-s + (−0.824 + 0.476i)4-s + (−0.182 + 0.119i)6-s + (−0.359 − 0.933i)7-s + (0.301 − 0.301i)8-s + (0.805 − 0.593i)9-s + (0.199 − 0.114i)11-s + (−0.634 + 0.709i)12-s + (0.629 + 0.629i)13-s + (0.128 + 0.176i)14-s + (0.429 − 0.744i)16-s + (0.305 − 1.13i)17-s + (−0.136 + 0.170i)18-s + (0.325 + 0.187i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.702 + 0.711i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.702 + 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.702 + 0.711i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (443, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.702 + 0.711i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.35925 - 0.568186i\)
\(L(\frac12)\) \(\approx\) \(1.35925 - 0.568186i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.64 + 0.540i)T \)
5 \( 1 \)
7 \( 1 + (0.951 + 2.46i)T \)
good2 \( 1 + (0.298 - 0.0799i)T + (1.73 - i)T^{2} \)
11 \( 1 + (-0.660 + 0.381i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.27 - 2.27i)T + 13iT^{2} \)
17 \( 1 + (-1.25 + 4.69i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-1.41 - 0.818i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.98 + 7.39i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 - 4.94T + 29T^{2} \)
31 \( 1 + (-2.96 - 5.13i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.915 - 3.41i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 4.35iT - 41T^{2} \)
43 \( 1 + (2.69 + 2.69i)T + 43iT^{2} \)
47 \( 1 + (4.14 - 1.10i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-6.71 - 1.79i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (3.84 + 6.65i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.19 - 3.80i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.0471 - 0.0126i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 12.4iT - 71T^{2} \)
73 \( 1 + (0.359 - 1.34i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (3.66 + 2.11i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.05 - 5.05i)T - 83iT^{2} \)
89 \( 1 + (-0.453 + 0.785i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.73 - 3.73i)T - 97iT^{2} \)
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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

−10.36993502511362991899355063912, −9.748800430618257356506063045279, −8.823904740943688003163922460765, −8.240952695326446371300620177729, −7.24202829416285210972213043955, −6.55524015346743219764586331780, −4.71541931464294786430598495858, −3.88695867047891119457663623636, −2.92256707772745846577156881596, −0.978723718878599032351967482496, 1.61331171939871849935229517215, 3.13241170685815560190479700174, 4.13340313850287218232743811787, 5.34176433849027544819001548289, 6.20566607397011043366370508262, 7.83873464710232491942824914047, 8.415872519916093118223312919172, 9.266666380742437076717264144782, 9.840544050364333350345616352133, 10.62034495200802455767566036715

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