Properties

Label 2-525-21.5-c1-0-22
Degree $2$
Conductor $525$
Sign $0.980 - 0.195i$
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  + (0.933 + 0.539i)2-s + (−0.918 + 1.46i)3-s + (−0.418 − 0.725i)4-s + (−1.64 + 0.876i)6-s + (2.47 − 0.929i)7-s − 3.05i·8-s + (−1.31 − 2.69i)9-s + (3.84 − 2.21i)11-s + (1.44 + 0.0513i)12-s + 0.955i·13-s + (2.81 + 0.467i)14-s + (0.812 − 1.40i)16-s + (0.253 + 0.439i)17-s + (0.228 − 3.22i)18-s + (4.41 + 2.54i)19-s + ⋯
L(s)  = 1  + (0.660 + 0.381i)2-s + (−0.530 + 0.847i)3-s + (−0.209 − 0.362i)4-s + (−0.673 + 0.357i)6-s + (0.936 − 0.351i)7-s − 1.08i·8-s + (−0.437 − 0.899i)9-s + (1.15 − 0.669i)11-s + (0.418 + 0.0148i)12-s + 0.265i·13-s + (0.752 + 0.125i)14-s + (0.203 − 0.351i)16-s + (0.0615 + 0.106i)17-s + (0.0539 − 0.760i)18-s + (1.01 + 0.584i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.195i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 - 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.980 - 0.195i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.980 - 0.195i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.78177 + 0.175806i\)
\(L(\frac12)\) \(\approx\) \(1.78177 + 0.175806i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.918 - 1.46i)T \)
5 \( 1 \)
7 \( 1 + (-2.47 + 0.929i)T \)
good2 \( 1 + (-0.933 - 0.539i)T + (1 + 1.73i)T^{2} \)
11 \( 1 + (-3.84 + 2.21i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 0.955iT - 13T^{2} \)
17 \( 1 + (-0.253 - 0.439i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.41 - 2.54i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.72 + 2.14i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 6.89iT - 29T^{2} \)
31 \( 1 + (-5.10 + 2.94i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.76 + 6.51i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 4.65T + 41T^{2} \)
43 \( 1 - 0.492T + 43T^{2} \)
47 \( 1 + (3.32 - 5.76i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-7.90 + 4.56i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.81 + 10.0i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.399 - 0.230i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.85 + 3.20i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 7.90iT - 71T^{2} \)
73 \( 1 + (-5.46 + 3.15i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.38 - 12.7i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 10.7T + 83T^{2} \)
89 \( 1 + (3.57 - 6.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 6.91iT - 97T^{2} \)
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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

−10.94901806631532827493779266469, −10.02014659958468271190968259543, −9.296537988561542350100520190966, −8.288341827093952651594950416921, −6.89599336504476137819617770360, −6.01793010950269037076169507958, −5.23055545042431075896471902122, −4.30342886838245540371873644678, −3.61046061590080223838468225238, −1.12925068644563031668758793463, 1.52743753794532917103418172493, 2.75301512932216308253460757893, 4.27270976770131666629179651383, 5.07875661149778585830279098504, 6.05555746861211054725416031483, 7.26368491918229091330819909780, 8.018956488922050123191284073854, 8.877943824036223721923155540836, 10.12924833727132990701403338864, 11.51466455307476294567240591370

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