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Properties

Label 2-525-105.53-c1-0-1
Degree $2$
Conductor $525$
Sign $0.618 + 0.785i$
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  + (−0.631 + 2.35i)2-s + (0.775 + 1.54i)3-s + (−3.42 − 1.97i)4-s + (−4.13 + 0.849i)6-s + (−1.82 − 1.91i)7-s + (3.36 − 3.36i)8-s + (−1.79 + 2.40i)9-s + (−3.08 − 1.77i)11-s + (0.406 − 6.83i)12-s + (−1.28 − 1.28i)13-s + (5.67 − 3.08i)14-s + (1.85 + 3.21i)16-s + (−2.95 + 0.792i)17-s + (−4.52 − 5.75i)18-s + (−0.331 + 0.191i)19-s + ⋯
L(s)  = 1  + (−0.446 + 1.66i)2-s + (0.447 + 0.894i)3-s + (−1.71 − 0.987i)4-s + (−1.68 + 0.346i)6-s + (−0.688 − 0.725i)7-s + (1.19 − 1.19i)8-s + (−0.599 + 0.800i)9-s + (−0.928 − 0.536i)11-s + (0.117 − 1.97i)12-s + (−0.356 − 0.356i)13-s + (1.51 − 0.823i)14-s + (0.463 + 0.803i)16-s + (−0.717 + 0.192i)17-s + (−1.06 − 1.35i)18-s + (−0.0761 + 0.0439i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.618 + 0.785i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.618 + 0.785i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.618 + 0.785i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (368, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.618 + 0.785i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.109096 - 0.0529864i\)
\(L(\frac12)\) \(\approx\) \(0.109096 - 0.0529864i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.775 - 1.54i)T \)
5 \( 1 \)
7 \( 1 + (1.82 + 1.91i)T \)
good2 \( 1 + (0.631 - 2.35i)T + (-1.73 - i)T^{2} \)
11 \( 1 + (3.08 + 1.77i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.28 + 1.28i)T + 13iT^{2} \)
17 \( 1 + (2.95 - 0.792i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (0.331 - 0.191i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.45 - 0.658i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + 5.51T + 29T^{2} \)
31 \( 1 + (-0.323 + 0.561i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.00 + 1.34i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 10.1iT - 41T^{2} \)
43 \( 1 + (-0.335 - 0.335i)T + 43iT^{2} \)
47 \( 1 + (0.751 - 2.80i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-0.815 - 3.04i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (3.81 - 6.60i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.45 + 9.45i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.31 - 12.3i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 3.06iT - 71T^{2} \)
73 \( 1 + (3.17 - 0.849i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-3.21 + 1.85i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.973 - 0.973i)T - 83iT^{2} \)
89 \( 1 + (1.51 + 2.63i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-10.3 + 10.3i)T - 97iT^{2} \)
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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

−11.14410286858219405007854247047, −10.34328049168180281491842029350, −9.590167997244704980882872104943, −8.833909925796744286622443423079, −7.968158344253273578116447182263, −7.28688145943682178814747411325, −6.17347338787274618031126746946, −5.27927616305670219657349456091, −4.33756212533615080752821265988, −3.02800626826058706281779612475, 0.07298692804108732244395651591, 1.95904161440365494149647904156, 2.61175104162187660805604800720, 3.65567603583956267133090184416, 5.15955943992108611361115654808, 6.60699957030912963668873977703, 7.63851457955164352607256923444, 8.784762117072923886923403660239, 9.202128047165582431631102690043, 10.12974991401174414914366945305

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