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Properties

Label 2-3536-884.259-c0-0-1
Degree $2$
Conductor $3536$
Sign $0.615 + 0.788i$
Analytic cond. $1.76469$
Root an. cond. $1.32841$
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 − i)7-s − i·9-s + (1 + i)11-s + 13-s − 17-s + 2i·19-s − i·25-s + (1 − i)29-s + (1 − i)31-s + i·49-s − 2i·53-s − 2i·59-s + (1 + i)61-s + (−1 + i)63-s − 2·67-s + ⋯
L(s)  = 1  + (−1 − i)7-s − i·9-s + (1 + i)11-s + 13-s − 17-s + 2i·19-s − i·25-s + (1 − i)29-s + (1 − i)31-s + i·49-s − 2i·53-s − 2i·59-s + (1 + i)61-s + (−1 + i)63-s − 2·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.615 + 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.615 + 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3536\)    =    \(2^{4} \cdot 13 \cdot 17\)
Sign: $0.615 + 0.788i$
Analytic conductor: \(1.76469\)
Root analytic conductor: \(1.32841\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3536} (2911, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3536,\ (\ :0),\ 0.615 + 0.788i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.187775391\)
\(L(\frac12)\) \(\approx\) \(1.187775391\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 - T \)
17 \( 1 + T \)
good3 \( 1 + iT^{2} \)
5 \( 1 + iT^{2} \)
7 \( 1 + (1 + i)T + iT^{2} \)
11 \( 1 + (-1 - i)T + iT^{2} \)
19 \( 1 - 2iT - T^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 + (-1 + i)T - iT^{2} \)
31 \( 1 + (-1 + i)T - iT^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 - iT^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + 2iT - T^{2} \)
59 \( 1 + 2iT - T^{2} \)
61 \( 1 + (-1 - i)T + iT^{2} \)
67 \( 1 + 2T + T^{2} \)
71 \( 1 + (-1 + i)T - iT^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 - iT^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + iT^{2} \)
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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

−8.611484971257220684116842494163, −7.963856458132843542932698588111, −6.90035814141313934601640436005, −6.38823335392720395420816650816, −6.09631755394290041808224388572, −4.46306739568791347571807286480, −3.96686661983875974918246245976, −3.40291255098762958949181952242, −2.00227493400263195666312800300, −0.801665297405876454117248875569, 1.24723628622675250331372453507, 2.66816658239487499097204177435, 3.10575950412958241612881178393, 4.27628129687568692453249302374, 5.12324477104670220022357630522, 5.97567080075240066847263332155, 6.57335736143883780069943817323, 7.17349093986786997641044405164, 8.490181777930309893868544829112, 8.871530187442271119421451294793

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