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Properties

Label 2-525-7.4-c1-0-23
Degree $2$
Conductor $525$
Sign $-0.386 + 0.922i$
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  + (0.5 − 0.866i)3-s + (1 − 1.73i)4-s + (−2.5 − 0.866i)7-s + (−0.499 − 0.866i)9-s + (−0.999 − 1.73i)12-s + 13-s + (−1.99 − 3.46i)16-s + (3 − 5.19i)17-s + (−2.5 − 4.33i)19-s + (−2 + 1.73i)21-s + (3 + 5.19i)23-s − 0.999·27-s + (−4 + 3.46i)28-s − 6·29-s + (−2.5 + 4.33i)31-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + (0.5 − 0.866i)4-s + (−0.944 − 0.327i)7-s + (−0.166 − 0.288i)9-s + (−0.288 − 0.499i)12-s + 0.277·13-s + (−0.499 − 0.866i)16-s + (0.727 − 1.26i)17-s + (−0.573 − 0.993i)19-s + (−0.436 + 0.377i)21-s + (0.625 + 1.08i)23-s − 0.192·27-s + (−0.755 + 0.654i)28-s − 1.11·29-s + (−0.449 + 0.777i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.386 + 0.922i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ -0.386 + 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.811477 - 1.21993i\)
\(L(\frac12)\) \(\approx\) \(0.811477 - 1.21993i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 \)
7 \( 1 + (2.5 + 0.866i)T \)
good2 \( 1 + (-1 + 1.73i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - T + 13T^{2} \)
17 \( 1 + (-3 + 5.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.5 + 4.33i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3 - 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + (2.5 - 4.33i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.5 + 6.06i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 12T + 41T^{2} \)
43 \( 1 - T + 43T^{2} \)
47 \( 1 + (-3 - 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3 + 5.19i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1 + 1.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.5 - 6.06i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + (-5.5 + 9.52i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.5 - 11.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 12T + 83T^{2} \)
89 \( 1 + (3 + 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 10T + 97T^{2} \)
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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

−10.70846813759650693748331923799, −9.429261614449865581078979072505, −9.235400865476015650543708733492, −7.51377088805424996764034962314, −7.02946053841575580833924233650, −6.06122092993716537186928837378, −5.13873784339039782419533359098, −3.51489354525652083088572285360, −2.39521455229571783979126256851, −0.825013239159957376630197659206, 2.23092526992372195950076853176, 3.42032938587321839439936855819, 4.06750183712284442245157543694, 5.74398997712566422463911571178, 6.52470981298681098802139282841, 7.69361923935022553387033424727, 8.468909266431393032577860483417, 9.288955076373747925903635132525, 10.33642590267498942210165486600, 10.98203419319383542529659843865

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