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Properties

Label 2-3525-705.422-c0-0-29
Degree $2$
Conductor $3525$
Sign $-0.241 - 0.970i$
Analytic cond. $1.75920$
Root an. cond. $1.32634$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.29 − 1.29i)2-s + (−0.544 − 0.838i)3-s − 2.33i·4-s + (−1.78 − 0.379i)6-s + (−1.40 − 1.40i)7-s + (−1.72 − 1.72i)8-s + (−0.406 + 0.913i)9-s + (−1.96 + 1.27i)12-s − 3.63·14-s − 2.12·16-s + (0.946 − 0.946i)17-s + (0.654 + 1.70i)18-s + (−0.413 + 1.94i)21-s + (−0.508 + 2.39i)24-s + (0.987 − 0.156i)27-s + (−3.28 + 3.28i)28-s + ⋯
L(s)  = 1  + (1.29 − 1.29i)2-s + (−0.544 − 0.838i)3-s − 2.33i·4-s + (−1.78 − 0.379i)6-s + (−1.40 − 1.40i)7-s + (−1.72 − 1.72i)8-s + (−0.406 + 0.913i)9-s + (−1.96 + 1.27i)12-s − 3.63·14-s − 2.12·16-s + (0.946 − 0.946i)17-s + (0.654 + 1.70i)18-s + (−0.413 + 1.94i)21-s + (−0.508 + 2.39i)24-s + (0.987 − 0.156i)27-s + (−3.28 + 3.28i)28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.241 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.241 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3525\)    =    \(3 \cdot 5^{2} \cdot 47\)
Sign: $-0.241 - 0.970i$
Analytic conductor: \(1.75920\)
Root analytic conductor: \(1.32634\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3525} (1832, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3525,\ (\ :0),\ -0.241 - 0.970i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.495819648\)
\(L(\frac12)\) \(\approx\) \(1.495819648\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.544 + 0.838i)T \)
5 \( 1 \)
47 \( 1 + (0.707 - 0.707i)T \)
good2 \( 1 + (-1.29 + 1.29i)T - iT^{2} \)
7 \( 1 + (1.40 + 1.40i)T + iT^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 + (-0.946 + 0.946i)T - iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (-1.34 - 1.34i)T + iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + iT^{2} \)
53 \( 1 + (1.14 + 1.14i)T + iT^{2} \)
59 \( 1 + 1.98T + T^{2} \)
61 \( 1 - 1.61T + T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 + 0.415iT - T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 - 0.618iT - T^{2} \)
83 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
89 \( 1 - 1.17T + T^{2} \)
97 \( 1 + (0.831 + 0.831i)T + iT^{2} \)
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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

−7.982831597365124665662646772463, −7.18225710749613661184640242464, −6.46214843285602524163946469409, −5.95693990202705281006955251905, −4.97202458508804711401906020551, −4.31733141482435182114774651378, −3.25216289129419603664744808270, −2.88666372955265454473260007271, −1.50910043356200853216441081026, −0.61387806307452345091688026962, 2.63638101871862822063697572217, 3.41223616781987969391763153520, 3.97436449278312138173201266580, 5.01207000213234208913297867883, 5.62394871785932660695228648570, 6.15476151930345223400734973166, 6.50051638550127204155698582540, 7.59291462999476219505736013549, 8.438898646026066809693188784848, 9.182706750869618950863814808377

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