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Properties

Label 2-525-35.17-c1-0-13
Degree $2$
Conductor $525$
Sign $0.690 + 0.723i$
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  + (0.124 − 0.464i)2-s + (−0.965 + 0.258i)3-s + (1.53 + 0.884i)4-s + 0.480i·6-s + (−1.13 − 2.38i)7-s + (1.28 − 1.28i)8-s + (0.866 − 0.499i)9-s + (1.38 − 2.39i)11-s + (−1.70 − 0.457i)12-s + (0.707 + 0.707i)13-s + (−1.25 + 0.231i)14-s + (1.33 + 2.30i)16-s + (−1.14 − 4.28i)17-s + (−0.124 − 0.464i)18-s + (−0.280 − 0.486i)19-s + ⋯
L(s)  = 1  + (0.0880 − 0.328i)2-s + (−0.557 + 0.149i)3-s + (0.765 + 0.442i)4-s + 0.196i·6-s + (−0.430 − 0.902i)7-s + (0.453 − 0.453i)8-s + (0.288 − 0.166i)9-s + (0.417 − 0.722i)11-s + (−0.493 − 0.132i)12-s + (0.196 + 0.196i)13-s + (−0.334 + 0.0618i)14-s + (0.333 + 0.577i)16-s + (−0.278 − 1.03i)17-s + (−0.0293 − 0.109i)18-s + (−0.0643 − 0.111i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.690 + 0.723i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.690 + 0.723i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.690 + 0.723i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.690 + 0.723i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.39470 - 0.596564i\)
\(L(\frac12)\) \(\approx\) \(1.39470 - 0.596564i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.965 - 0.258i)T \)
5 \( 1 \)
7 \( 1 + (1.13 + 2.38i)T \)
good2 \( 1 + (-0.124 + 0.464i)T + (-1.73 - i)T^{2} \)
11 \( 1 + (-1.38 + 2.39i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.707 - 0.707i)T + 13iT^{2} \)
17 \( 1 + (1.14 + 4.28i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (0.280 + 0.486i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-7.77 - 2.08i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + 5.53iT - 29T^{2} \)
31 \( 1 + (-6.97 - 4.02i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.09 - 4.09i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + 4.21iT - 41T^{2} \)
43 \( 1 + (-1.08 + 1.08i)T - 43iT^{2} \)
47 \( 1 + (8.46 + 2.26i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-0.194 - 0.727i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-5.09 + 8.81i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.5 - 4.33i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-9.65 + 2.58i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 14.4T + 71T^{2} \)
73 \( 1 + (7.60 - 2.03i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-8.05 + 4.65i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6.20 - 6.20i)T + 83iT^{2} \)
89 \( 1 + (-7.67 - 13.2i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.41 - 6.41i)T - 97iT^{2} \)
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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

−10.93489630970651059901405976098, −10.12129620075117170605361509927, −9.134743603108980552019499317514, −7.899578185155338761882213318708, −6.86073441533332534997910297469, −6.45458364796594804570890938293, −4.97994493584559310792081668766, −3.82800569077864091711754068909, −2.89008526245694080810980298844, −1.04241162680889422622471840881, 1.56629606871687372685580038538, 2.87661225136349458431603224017, 4.59887354134091536702220788418, 5.61362730014660165743152943252, 6.39967974934758827696273034343, 7.00859433185883322935625770447, 8.199047494274310535135524832286, 9.254818973986090943708083031647, 10.24303297779744206845543292020, 10.99582204195181801320390133365

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