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Properties

Label 2-525-35.34-c2-0-44
Degree $2$
Conductor $525$
Sign $-0.285 - 0.958i$
Analytic cond. $14.3052$
Root an. cond. $3.78222$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.26i·2-s + 1.73·3-s − 6.64·4-s − 5.65i·6-s + (−4.78 − 5.10i)7-s + 8.63i·8-s + 2.99·9-s + 14.4·11-s − 11.5·12-s − 14.5·13-s + (−16.6 + 15.6i)14-s + 1.58·16-s − 32.4·17-s − 9.78i·18-s − 5.24i·19-s + ⋯
L(s)  = 1  − 1.63i·2-s + 0.577·3-s − 1.66·4-s − 0.941i·6-s + (−0.683 − 0.729i)7-s + 1.07i·8-s + 0.333·9-s + 1.31·11-s − 0.959·12-s − 1.12·13-s + (−1.19 + 1.11i)14-s + 0.0988·16-s − 1.91·17-s − 0.543i·18-s − 0.276i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.285 - 0.958i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.285 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.285 - 0.958i$
Analytic conductor: \(14.3052\)
Root analytic conductor: \(3.78222\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1),\ -0.285 - 0.958i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.9088822085\)
\(L(\frac12)\) \(\approx\) \(0.9088822085\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 1.73T \)
5 \( 1 \)
7 \( 1 + (4.78 + 5.10i)T \)
good2 \( 1 + 3.26iT - 4T^{2} \)
11 \( 1 - 14.4T + 121T^{2} \)
13 \( 1 + 14.5T + 169T^{2} \)
17 \( 1 + 32.4T + 289T^{2} \)
19 \( 1 + 5.24iT - 361T^{2} \)
23 \( 1 + 11.0iT - 529T^{2} \)
29 \( 1 + 23.8T + 841T^{2} \)
31 \( 1 + 12.9iT - 961T^{2} \)
37 \( 1 + 1.35iT - 1.36e3T^{2} \)
41 \( 1 - 26.3iT - 1.68e3T^{2} \)
43 \( 1 + 32.2iT - 1.84e3T^{2} \)
47 \( 1 - 29.2T + 2.20e3T^{2} \)
53 \( 1 - 37.7iT - 2.80e3T^{2} \)
59 \( 1 + 104. iT - 3.48e3T^{2} \)
61 \( 1 - 106. iT - 3.72e3T^{2} \)
67 \( 1 + 72.2iT - 4.48e3T^{2} \)
71 \( 1 + 100.T + 5.04e3T^{2} \)
73 \( 1 + 20.3T + 5.32e3T^{2} \)
79 \( 1 - 125.T + 6.24e3T^{2} \)
83 \( 1 + 53.1T + 6.88e3T^{2} \)
89 \( 1 + 163. iT - 7.92e3T^{2} \)
97 \( 1 + 80.8T + 9.40e3T^{2} \)
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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

−10.08121508889758564436007263645, −9.307357992858884876550946610669, −8.862691755040950667397246006748, −7.30896123745794273078390153918, −6.50219014509583105624703118929, −4.53621333701231645857394082674, −3.98788395768134079801468680050, −2.86728881097439869690091775133, −1.85073191811617811810006395236, −0.31528373393825391011737454067, 2.27963720326015011022594783282, 3.86807073886686378375723086655, 4.90056140614569012420238495087, 6.05261431749895876074337091076, 6.78014678771784266187376968594, 7.45683635508858756647146384245, 8.693552952622222713459777528384, 9.078113255930300641311011131403, 9.806414653470330306420390177669, 11.34809618844437902796936885403

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