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Properties

Label 2-525-15.2-c1-0-6
Degree $2$
Conductor $525$
Sign $-0.327 - 0.944i$
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  + (0.347 − 0.347i)2-s + (−1.72 + 0.176i)3-s + 1.75i·4-s + (−0.536 + 0.659i)6-s + (0.707 + 0.707i)7-s + (1.30 + 1.30i)8-s + (2.93 − 0.607i)9-s − 2.67i·11-s + (−0.310 − 3.03i)12-s + (−2.14 + 2.14i)13-s + 0.490·14-s − 2.61·16-s + (−3.26 + 3.26i)17-s + (0.808 − 1.23i)18-s + 5.24i·19-s + ⋯
L(s)  = 1  + (0.245 − 0.245i)2-s + (−0.994 + 0.101i)3-s + 0.879i·4-s + (−0.219 + 0.269i)6-s + (0.267 + 0.267i)7-s + (0.461 + 0.461i)8-s + (0.979 − 0.202i)9-s − 0.805i·11-s + (−0.0895 − 0.874i)12-s + (−0.596 + 0.596i)13-s + 0.131·14-s − 0.653·16-s + (−0.792 + 0.792i)17-s + (0.190 − 0.290i)18-s + 1.20i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.327 - 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.327 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.327 - 0.944i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (407, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ -0.327 - 0.944i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.533244 + 0.749347i\)
\(L(\frac12)\) \(\approx\) \(0.533244 + 0.749347i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.72 - 0.176i)T \)
5 \( 1 \)
7 \( 1 + (-0.707 - 0.707i)T \)
good2 \( 1 + (-0.347 + 0.347i)T - 2iT^{2} \)
11 \( 1 + 2.67iT - 11T^{2} \)
13 \( 1 + (2.14 - 2.14i)T - 13iT^{2} \)
17 \( 1 + (3.26 - 3.26i)T - 17iT^{2} \)
19 \( 1 - 5.24iT - 19T^{2} \)
23 \( 1 + (-2.54 - 2.54i)T + 23iT^{2} \)
29 \( 1 + 2.86T + 29T^{2} \)
31 \( 1 + 5.28T + 31T^{2} \)
37 \( 1 + (-2.14 - 2.14i)T + 37iT^{2} \)
41 \( 1 - 11.5iT - 41T^{2} \)
43 \( 1 + (0.759 - 0.759i)T - 43iT^{2} \)
47 \( 1 + (-7.66 + 7.66i)T - 47iT^{2} \)
53 \( 1 + (4.43 + 4.43i)T + 53iT^{2} \)
59 \( 1 - 0.159T + 59T^{2} \)
61 \( 1 - 4.72T + 61T^{2} \)
67 \( 1 + (-5.41 - 5.41i)T + 67iT^{2} \)
71 \( 1 + 13.5iT - 71T^{2} \)
73 \( 1 + (4.16 - 4.16i)T - 73iT^{2} \)
79 \( 1 - 3.89iT - 79T^{2} \)
83 \( 1 + (4.03 + 4.03i)T + 83iT^{2} \)
89 \( 1 - 3.95T + 89T^{2} \)
97 \( 1 + (-1.86 - 1.86i)T + 97iT^{2} \)
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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

−11.38975940291679865179203970645, −10.54864645389173696611523755115, −9.408603334460123404323237445435, −8.405236559073647535749115670175, −7.48134438462922091044711635831, −6.47532943887545781223269743361, −5.46808228550363965027983182695, −4.43576385247174634111076444054, −3.51131354909262729064437999301, −1.87622861357476000252216729477, 0.55918542433165310246539073764, 2.18484985907224781962423762043, 4.37063446287184218478206340336, 4.98762238199600033402808876798, 5.81159669442540735443873759371, 7.09217497252076894725006834186, 7.23132056784694255127824634204, 9.082607893650920605144651279795, 9.844830697414123543057746600438, 10.81742048149702737117165560415

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