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Properties

Label 2-525-35.33-c1-0-10
Degree $2$
Conductor $525$
Sign $0.611 - 0.791i$
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  + (0.173 + 0.648i)2-s + (0.965 + 0.258i)3-s + (1.34 − 0.774i)4-s + 0.671i·6-s + (0.588 + 2.57i)7-s + (1.68 + 1.68i)8-s + (0.866 + 0.499i)9-s + (0.0701 + 0.121i)11-s + (1.49 − 0.401i)12-s + (−2.35 + 2.35i)13-s + (−1.56 + 0.829i)14-s + (0.750 − 1.29i)16-s + (1.97 − 7.37i)17-s + (−0.173 + 0.648i)18-s + (−3.89 + 6.74i)19-s + ⋯
L(s)  = 1  + (0.122 + 0.458i)2-s + (0.557 + 0.149i)3-s + (0.670 − 0.387i)4-s + 0.273i·6-s + (0.222 + 0.974i)7-s + (0.595 + 0.595i)8-s + (0.288 + 0.166i)9-s + (0.0211 + 0.0366i)11-s + (0.432 − 0.115i)12-s + (−0.653 + 0.653i)13-s + (−0.419 + 0.221i)14-s + (0.187 − 0.324i)16-s + (0.478 − 1.78i)17-s + (−0.0409 + 0.152i)18-s + (−0.893 + 1.54i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.611 - 0.791i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.611 - 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.611 - 0.791i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (418, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.611 - 0.791i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.99383 + 0.979244i\)
\(L(\frac12)\) \(\approx\) \(1.99383 + 0.979244i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.588 - 2.57i)T \)
good2 \( 1 + (-0.173 - 0.648i)T + (-1.73 + i)T^{2} \)
11 \( 1 + (-0.0701 - 0.121i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.35 - 2.35i)T - 13iT^{2} \)
17 \( 1 + (-1.97 + 7.37i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (3.89 - 6.74i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.50 + 0.671i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 5.09iT - 29T^{2} \)
31 \( 1 + (-2.54 + 1.46i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.53 + 5.73i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 0.261iT - 41T^{2} \)
43 \( 1 + (-2.11 - 2.11i)T + 43iT^{2} \)
47 \( 1 + (1.50 - 0.402i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (0.749 - 2.79i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (4.37 + 7.57i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.76 + 2.75i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.54 + 2.02i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 3.56T + 71T^{2} \)
73 \( 1 + (-3.16 - 0.847i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-0.113 - 0.0656i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (7.33 - 7.33i)T - 83iT^{2} \)
89 \( 1 + (-2.44 + 4.23i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.25 - 1.25i)T + 97iT^{2} \)
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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

−11.03488828144105699756460715001, −9.915686048289691988719272144161, −9.279235802655680642872046529747, −8.145534130897693458398950490270, −7.43036480134847808947658260746, −6.39916821626615084797523259529, −5.46512441622220282407481823476, −4.51420654332259584816194969961, −2.83860523127698005700222716600, −1.92618776590651694970214506673, 1.40940026215608994300683522482, 2.75138870478693749758718750314, 3.70771092657296127893348877186, 4.75284935144317813815009685164, 6.40776271158942461118263533297, 7.20834261555293586406088204264, 7.958163466751086586174590011101, 8.846925087396314878132958083303, 10.30378716225681434930492193156, 10.55009445223316886291365203901

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