Properties

Label 2-56e2-28.27-c1-0-13
Degree $2$
Conductor $3136$
Sign $0.755 - 0.654i$
Analytic cond. $25.0410$
Root an. cond. $5.00410$
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  − 2.64·3-s − 1.73i·5-s + 4.00·9-s − 4.58i·11-s + 3.46i·13-s + 4.58i·15-s + 5.19i·17-s − 2.64·19-s − 4.58i·23-s + 2.00·25-s − 2.64·27-s − 2.64·31-s + 12.1i·33-s − 7·37-s − 9.16i·39-s + ⋯
L(s)  = 1  − 1.52·3-s − 0.774i·5-s + 1.33·9-s − 1.38i·11-s + 0.960i·13-s + 1.18i·15-s + 1.26i·17-s − 0.606·19-s − 0.955i·23-s + 0.400·25-s − 0.509·27-s − 0.475·31-s + 2.11i·33-s − 1.15·37-s − 1.46i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3136\)    =    \(2^{6} \cdot 7^{2}\)
Sign: $0.755 - 0.654i$
Analytic conductor: \(25.0410\)
Root analytic conductor: \(5.00410\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3136} (3135, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3136,\ (\ :1/2),\ 0.755 - 0.654i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6413188053\)
\(L(\frac12)\) \(\approx\) \(0.6413188053\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + 2.64T + 3T^{2} \)
5 \( 1 + 1.73iT - 5T^{2} \)
11 \( 1 + 4.58iT - 11T^{2} \)
13 \( 1 - 3.46iT - 13T^{2} \)
17 \( 1 - 5.19iT - 17T^{2} \)
19 \( 1 + 2.64T + 19T^{2} \)
23 \( 1 + 4.58iT - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 2.64T + 31T^{2} \)
37 \( 1 + 7T + 37T^{2} \)
41 \( 1 - 3.46iT - 41T^{2} \)
43 \( 1 - 9.16iT - 43T^{2} \)
47 \( 1 + 7.93T + 47T^{2} \)
53 \( 1 + 3T + 53T^{2} \)
59 \( 1 - 7.93T + 59T^{2} \)
61 \( 1 - 1.73iT - 61T^{2} \)
67 \( 1 - 4.58iT - 67T^{2} \)
71 \( 1 + 9.16iT - 71T^{2} \)
73 \( 1 + 5.19iT - 73T^{2} \)
79 \( 1 + 4.58iT - 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + 1.73iT - 89T^{2} \)
97 \( 1 - 3.46iT - 97T^{2} \)
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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

−8.603959685117605274624274799275, −8.281390907206189276589714129924, −6.95873441802458218855721305703, −6.26425990786733872315462468550, −5.89381048525272325134644529847, −4.92099515484713832889273029018, −4.42283027754529808553492740985, −3.37918003264812349990272047166, −1.78507192671630068897529129916, −0.78695305702569501171158052764, 0.35230119539544525332270678223, 1.79966140211729583526849501064, 2.94929372254677693193953866936, 4.03869918000130761606394770362, 5.14314880454126610952100331277, 5.32539454172096132771650614404, 6.41825661967555690090194121871, 7.08363225196803871112505006140, 7.38198533927081371147750091926, 8.593323387202409771864500654559

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