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Properties

Label 2-525-105.23-c1-0-26
Degree $2$
Conductor $525$
Sign $0.263 + 0.964i$
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  + (−0.448 + 1.67i)3-s + (−1.73 + i)4-s + (−0.189 − 2.63i)7-s + (−2.59 − 1.50i)9-s + (−0.896 − 3.34i)12-s + (−4.89 − 4.89i)13-s + (1.99 − 3.46i)16-s + (6.92 + 4i)19-s + (4.50 + 0.866i)21-s + (3.67 − 3.67i)27-s + (2.96 + 4.38i)28-s + (−3.5 − 6.06i)31-s + 6·36-s + (−1.34 − 5.01i)37-s + (10.3 − 6i)39-s + ⋯
L(s)  = 1  + (−0.258 + 0.965i)3-s + (−0.866 + 0.5i)4-s + (−0.0716 − 0.997i)7-s + (−0.866 − 0.5i)9-s + (−0.258 − 0.965i)12-s + (−1.35 − 1.35i)13-s + (0.499 − 0.866i)16-s + (1.58 + 0.917i)19-s + (0.981 + 0.188i)21-s + (0.707 − 0.707i)27-s + (0.560 + 0.827i)28-s + (−0.628 − 1.08i)31-s + 36-s + (−0.221 − 0.825i)37-s + (1.66 − 0.960i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.263 + 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.263 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.263 + 0.964i$
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: \((2,\ 525,\ (\ :1/2),\ 0.263 + 0.964i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.419014 - 0.320060i\)
\(L(\frac12)\) \(\approx\) \(0.419014 - 0.320060i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.448 - 1.67i)T \)
5 \( 1 \)
7 \( 1 + (0.189 + 2.63i)T \)
good2 \( 1 + (1.73 - i)T^{2} \)
11 \( 1 + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (4.89 + 4.89i)T + 13iT^{2} \)
17 \( 1 + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-6.92 - 4i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + (3.5 + 6.06i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.34 + 5.01i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + (8.57 + 8.57i)T + 43iT^{2} \)
47 \( 1 + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.34 + 0.896i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (-0.448 + 1.67i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (14.7 + 8.5i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 83iT^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.67 - 3.67i)T - 97iT^{2} \)
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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

−10.25213556861927504923765303053, −10.01181696984470297130345573559, −9.116791632197171407802292718576, −7.928910789817627855487166605169, −7.35268553143001976410536326528, −5.61980306807301087318875464467, −4.96809413726649574342654001295, −3.88890130540430168255199223667, −3.13923667701264943910482808242, −0.33351506282151773829195044777, 1.56807672724300909126568975963, 2.91069122966161678284892715467, 4.80528131891132812416345878566, 5.34326606652168598403536128198, 6.49546085631763092620316796590, 7.32561133558246440372880892047, 8.495245673572946280267123051348, 9.241594365070928334023458415877, 9.918774270014181691404913199681, 11.35690415063380749659196078123

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