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Properties

Label 2-56e2-112.5-c0-0-0
Degree $2$
Conductor $3136$
Sign $0.946 + 0.323i$
Analytic cond. $1.56506$
Root an. cond. $1.25102$
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  + (0.866 + 0.5i)9-s + (0.366 − 1.36i)11-s + (0.866 − 0.5i)25-s + (−1 + i)29-s + (1.36 − 0.366i)37-s + (−1 − i)43-s + (0.366 − 1.36i)53-s + (1.36 + 0.366i)67-s + 2i·71-s + (0.499 + 0.866i)81-s + (1 − 0.999i)99-s + (1.36 − 0.366i)107-s + (−1.36 − 0.366i)109-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)9-s + (0.366 − 1.36i)11-s + (0.866 − 0.5i)25-s + (−1 + i)29-s + (1.36 − 0.366i)37-s + (−1 − i)43-s + (0.366 − 1.36i)53-s + (1.36 + 0.366i)67-s + 2i·71-s + (0.499 + 0.866i)81-s + (1 − 0.999i)99-s + (1.36 − 0.366i)107-s + (−1.36 − 0.366i)109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.946 + 0.323i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.946 + 0.323i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3136\)    =    \(2^{6} \cdot 7^{2}\)
Sign: $0.946 + 0.323i$
Analytic conductor: \(1.56506\)
Root analytic conductor: \(1.25102\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3136} (3057, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3136,\ (\ :0),\ 0.946 + 0.323i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.378724920\)
\(L(\frac12)\) \(\approx\) \(1.378724920\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.866 - 0.5i)T^{2} \)
5 \( 1 + (-0.866 + 0.5i)T^{2} \)
11 \( 1 + (-0.366 + 1.36i)T + (-0.866 - 0.5i)T^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.866 - 0.5i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (1 - i)T - iT^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (1 + i)T + iT^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (-0.366 + 1.36i)T + (-0.866 - 0.5i)T^{2} \)
59 \( 1 + (0.866 + 0.5i)T^{2} \)
61 \( 1 + (0.866 - 0.5i)T^{2} \)
67 \( 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2} \)
71 \( 1 - 2iT - T^{2} \)
73 \( 1 + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + (-0.5 - 0.866i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−8.682250208655646768526070658603, −8.233400399758503863782411449716, −7.24085081406131317289457728582, −6.68486403424704279045783132162, −5.74342495686621207165181325713, −5.04930944370244658403716460483, −4.06578530106856223698586506530, −3.33736286905898660854639918227, −2.21529179421811501306765966677, −1.02982319841302847477671770980, 1.26526275253830947253151746897, 2.24242673102949340243586970179, 3.43813523235195492319251795889, 4.33672081642617143932404451367, 4.85029370982858669751953674137, 6.01947216857468067457112326503, 6.72568165614184667616974667195, 7.38376213643559866521726452997, 8.004719407124353915476593972474, 9.165226129890329951948537528480

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