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Properties

Label 2-525-15.8-c1-0-11
Degree $2$
Conductor $525$
Sign $-0.317 - 0.948i$
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  + (1.78 + 1.78i)2-s + (−0.912 − 1.47i)3-s + 4.37i·4-s + (1 − 4.25i)6-s + (0.707 − 0.707i)7-s + (−4.23 + 4.23i)8-s + (−1.33 + 2.68i)9-s + 5.84i·11-s + (6.43 − 3.98i)12-s + (2.38 + 2.38i)13-s + 2.52·14-s − 6.37·16-s + (3.00 + 3.00i)17-s + (−7.17 + 2.41i)18-s − 4i·19-s + ⋯
L(s)  = 1  + (1.26 + 1.26i)2-s + (−0.526 − 0.850i)3-s + 2.18i·4-s + (0.408 − 1.73i)6-s + (0.267 − 0.267i)7-s + (−1.49 + 1.49i)8-s + (−0.445 + 0.895i)9-s + 1.76i·11-s + (1.85 − 1.15i)12-s + (0.661 + 0.661i)13-s + 0.674·14-s − 1.59·16-s + (0.729 + 0.729i)17-s + (−1.69 + 0.568i)18-s − 0.917i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.317 - 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.317 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.317 - 0.948i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (218, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ -0.317 - 0.948i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.36575 + 1.89702i\)
\(L(\frac12)\) \(\approx\) \(1.36575 + 1.89702i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.912 + 1.47i)T \)
5 \( 1 \)
7 \( 1 + (-0.707 + 0.707i)T \)
good2 \( 1 + (-1.78 - 1.78i)T + 2iT^{2} \)
11 \( 1 - 5.84iT - 11T^{2} \)
13 \( 1 + (-2.38 - 2.38i)T + 13iT^{2} \)
17 \( 1 + (-3.00 - 3.00i)T + 17iT^{2} \)
19 \( 1 + 4iT - 19T^{2} \)
23 \( 1 + (-2.44 + 2.44i)T - 23iT^{2} \)
29 \( 1 + 2.67T + 29T^{2} \)
31 \( 1 + 6.74T + 31T^{2} \)
37 \( 1 + (-4.76 + 4.76i)T - 37iT^{2} \)
41 \( 1 + 5.34iT - 41T^{2} \)
43 \( 1 + (5.65 + 5.65i)T + 43iT^{2} \)
47 \( 1 + (5.45 + 5.45i)T + 47iT^{2} \)
53 \( 1 + (-3.77 + 3.77i)T - 53iT^{2} \)
59 \( 1 - 5.34T + 59T^{2} \)
61 \( 1 + 4.74T + 61T^{2} \)
67 \( 1 + (0.887 - 0.887i)T - 67iT^{2} \)
71 \( 1 - 8.51iT - 71T^{2} \)
73 \( 1 + (0.887 + 0.887i)T + 73iT^{2} \)
79 \( 1 + 2.11iT - 79T^{2} \)
83 \( 1 + (-6.93 + 6.93i)T - 83iT^{2} \)
89 \( 1 - 17.0T + 89T^{2} \)
97 \( 1 + (2.38 - 2.38i)T - 97iT^{2} \)
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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

−11.56834335501047114404997694471, −10.47403744848828816415334318515, −8.957393376065834947567702788433, −7.86671888906672155412675368237, −7.12746303337343219104079066263, −6.67800058931861506612730100184, −5.56746468472817109818589312122, −4.80901389099778392458022377054, −3.81818593848318653111686104213, −2.01716724423702025216934261765, 1.09728175084501692546911964700, 3.14450253991519253544179221408, 3.50823412956914362758717167069, 4.81537340324786388296976602745, 5.66200150070201798157233298638, 6.09141347981274723657599713378, 8.109293084037312469959973345051, 9.277031849549020593383752940853, 10.10796691315187967832039640873, 11.09563168419509300952391833027

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