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Properties

Label 2-3525-705.203-c0-0-1
Degree $2$
Conductor $3525$
Sign $-0.991 - 0.127i$
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  + (−0.202 − 1.97i)2-s + (0.930 − 0.366i)3-s + (−2.86 + 0.595i)4-s + (−0.911 − 1.75i)6-s + (1.15 + 3.63i)8-s + (0.730 − 0.682i)9-s + (−2.44 + 1.60i)12-s + (4.25 − 1.84i)16-s + (0.113 − 0.660i)17-s + (−1.49 − 1.30i)18-s + (1.93 + 0.266i)19-s + (−1.25 − 0.129i)23-s + (2.40 + 2.95i)24-s + (0.429 − 0.903i)27-s + (−0.650 − 1.49i)31-s + (−2.63 − 4.67i)32-s + ⋯
L(s)  = 1  + (−0.202 − 1.97i)2-s + (0.930 − 0.366i)3-s + (−2.86 + 0.595i)4-s + (−0.911 − 1.75i)6-s + (1.15 + 3.63i)8-s + (0.730 − 0.682i)9-s + (−2.44 + 1.60i)12-s + (4.25 − 1.84i)16-s + (0.113 − 0.660i)17-s + (−1.49 − 1.30i)18-s + (1.93 + 0.266i)19-s + (−1.25 − 0.129i)23-s + (2.40 + 2.95i)24-s + (0.429 − 0.903i)27-s + (−0.650 − 1.49i)31-s + (−2.63 − 4.67i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.127i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.127i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3525\)    =    \(3 \cdot 5^{2} \cdot 47\)
Sign: $-0.991 - 0.127i$
Analytic conductor: \(1.75920\)
Root analytic conductor: \(1.32634\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3525} (2318, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3525,\ (\ :0),\ -0.991 - 0.127i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.317419979\)
\(L(\frac12)\) \(\approx\) \(1.317419979\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.930 + 0.366i)T \)
5 \( 1 \)
47 \( 1 + (-0.971 - 0.236i)T \)
good2 \( 1 + (0.202 + 1.97i)T + (-0.979 + 0.203i)T^{2} \)
7 \( 1 + (-0.631 + 0.775i)T^{2} \)
11 \( 1 + (-0.334 - 0.942i)T^{2} \)
13 \( 1 + (-0.136 + 0.990i)T^{2} \)
17 \( 1 + (-0.113 + 0.660i)T + (-0.942 - 0.334i)T^{2} \)
19 \( 1 + (-1.93 - 0.266i)T + (0.962 + 0.269i)T^{2} \)
23 \( 1 + (1.25 + 0.129i)T + (0.979 + 0.203i)T^{2} \)
29 \( 1 + (0.990 - 0.136i)T^{2} \)
31 \( 1 + (0.650 + 1.49i)T + (-0.682 + 0.730i)T^{2} \)
37 \( 1 + (0.887 - 0.460i)T^{2} \)
41 \( 1 + (-0.576 + 0.816i)T^{2} \)
43 \( 1 + (0.398 + 0.917i)T^{2} \)
53 \( 1 + (1.30 + 0.412i)T + (0.816 + 0.576i)T^{2} \)
59 \( 1 + (-0.917 - 0.398i)T^{2} \)
61 \( 1 + (-0.116 + 0.0709i)T + (0.460 - 0.887i)T^{2} \)
67 \( 1 + (-0.631 - 0.775i)T^{2} \)
71 \( 1 + (-0.203 + 0.979i)T^{2} \)
73 \( 1 + (-0.997 + 0.0682i)T^{2} \)
79 \( 1 + (0.311 - 1.11i)T + (-0.854 - 0.519i)T^{2} \)
83 \( 1 + (0.156 + 0.906i)T + (-0.942 + 0.334i)T^{2} \)
89 \( 1 + (0.962 - 0.269i)T^{2} \)
97 \( 1 + (0.730 - 0.682i)T^{2} \)
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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

−8.677934442019818157349546058715, −7.80864742611659709866425783378, −7.46276559935979794285775462482, −5.86440707078089739995050939996, −4.94276918253987231948439560636, −3.98149630624217510209734174159, −3.44064458494988006382456702321, −2.61509686379291449765813028858, −1.89132485255445957127855616168, −0.855616487072612586348028950117, 1.40465802397650586363668152596, 3.22424693188238934101139750420, 3.95410903330851657066009712640, 4.79105332500202139447308323094, 5.48169416065796731749910574939, 6.21358427065696620228469172352, 7.30149478659866573378992133655, 7.48058463073467751073514315632, 8.322170064107733786605001996611, 8.849578865356014789653406003523

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