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Properties

Label 2-3528-504.67-c0-0-9
Degree $2$
Conductor $3528$
Sign $-0.962 - 0.272i$
Analytic cond. $1.76070$
Root an. cond. $1.32691$
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  + (0.5 − 0.866i)2-s + (0.707 − 0.707i)3-s + (−0.499 − 0.866i)4-s + (−0.258 − 0.965i)6-s − 0.999·8-s − 1.00i·9-s − 1.73·11-s + (−0.965 − 0.258i)12-s + (−0.5 + 0.866i)16-s + (0.258 − 0.448i)17-s + (−0.866 − 0.500i)18-s + (−0.965 − 1.67i)19-s + (−0.866 + 1.49i)22-s + (−0.707 + 0.707i)24-s + 25-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (0.707 − 0.707i)3-s + (−0.499 − 0.866i)4-s + (−0.258 − 0.965i)6-s − 0.999·8-s − 1.00i·9-s − 1.73·11-s + (−0.965 − 0.258i)12-s + (−0.5 + 0.866i)16-s + (0.258 − 0.448i)17-s + (−0.866 − 0.500i)18-s + (−0.965 − 1.67i)19-s + (−0.866 + 1.49i)22-s + (−0.707 + 0.707i)24-s + 25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.962 - 0.272i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.962 - 0.272i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.962 - 0.272i$
Analytic conductor: \(1.76070\)
Root analytic conductor: \(1.32691\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3528} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :0),\ -0.962 - 0.272i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.417645952\)
\(L(\frac12)\) \(\approx\) \(1.417645952\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (-0.707 + 0.707i)T \)
7 \( 1 \)
good5 \( 1 - T^{2} \)
11 \( 1 + 1.73T + T^{2} \)
13 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 + (-0.258 + 0.448i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.965 + 1.67i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.258 - 0.448i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.965 + 1.67i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (-0.965 + 1.67i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (-0.258 - 0.448i)T + (-0.5 + 0.866i)T^{2} \)
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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

−8.379514660505394694656986996312, −7.73307109725191233318615145917, −6.81329053999741369420666507326, −6.11055946683921448661024495027, −5.03130923314159004980148983519, −4.55813228784058033478003077797, −3.20360498733543133815547006951, −2.76236470512407913308326455598, −1.99290318564691387889049396951, −0.59682731085724558934234827916, 2.19487534438092709080542231404, 3.04154564988119759426859105337, 3.87526035425631563472533750549, 4.57508267526902485963812662902, 5.45228937303084828859561180865, 5.88450034569608601041087680584, 7.12054190596327903756991620436, 7.72920395365650266553315929440, 8.404519386551632394484201813271, 8.745238408206858040587345872091

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