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Properties

Label 2-56e2-8.5-c1-0-12
Degree $2$
Conductor $3136$
Sign $-0.707 + 0.707i$
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  + 2i·3-s + 4i·5-s − 9-s + 2i·11-s + 4i·13-s − 8·15-s − 2·17-s − 6i·19-s − 11·25-s + 4i·27-s + 8i·29-s + 8·31-s − 4·33-s − 8i·37-s − 8·39-s + ⋯
L(s)  = 1  + 1.15i·3-s + 1.78i·5-s − 0.333·9-s + 0.603i·11-s + 1.10i·13-s − 2.06·15-s − 0.485·17-s − 1.37i·19-s − 2.20·25-s + 0.769i·27-s + 1.48i·29-s + 1.43·31-s − 0.696·33-s − 1.31i·37-s − 1.28·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3136\)    =    \(2^{6} \cdot 7^{2}\)
Sign: $-0.707 + 0.707i$
Analytic conductor: \(25.0410\)
Root analytic conductor: \(5.00410\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3136} (1569, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3136,\ (\ :1/2),\ -0.707 + 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.367265787\)
\(L(\frac12)\) \(\approx\) \(1.367265787\)
\(L(\frac{3}{2})\) 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 \( 1 - 2iT - 3T^{2} \)
5 \( 1 - 4iT - 5T^{2} \)
11 \( 1 - 2iT - 11T^{2} \)
13 \( 1 - 4iT - 13T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 + 6iT - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 8iT - 29T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 + 8iT - 37T^{2} \)
41 \( 1 + 10T + 41T^{2} \)
43 \( 1 - 2iT - 43T^{2} \)
47 \( 1 + 8T + 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 - 10iT - 59T^{2} \)
61 \( 1 + 4iT - 61T^{2} \)
67 \( 1 - 2iT - 67T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 - 6T + 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + 6iT - 83T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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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

−9.337740345186374845700511101938, −8.625126234669142390965667685909, −7.37465255585159860174696195791, −6.86135303238179194493046585279, −6.39972437515509583460445999502, −5.09887299327169344857220318140, −4.46388080741141509418774897393, −3.62826074722234604933079682280, −2.88342213822445727805302303955, −1.96613803356982411648777077716, 0.44147507189050015585369494216, 1.21767964594959544451573829636, 2.08519459010180559264408081605, 3.40653435768586061531216852975, 4.45767005383746075494024577986, 5.21003215360428475014748173103, 6.00231656089656798838812285196, 6.59266702105441550867394730044, 7.86059166450488347634678335551, 8.192702098117068456068239506218

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