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Properties

Label 2-637-91.51-c1-0-32
Degree $2$
Conductor $637$
Sign $-0.346 - 0.938i$
Analytic cond. $5.08647$
Root an. cond. $2.25532$
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  + (2.14 + 1.24i)2-s + (0.837 + 1.45i)3-s + (2.07 + 3.59i)4-s + (0.584 + 0.337i)5-s + 4.15i·6-s + 5.35i·8-s + (0.0969 − 0.167i)9-s + (0.837 + 1.45i)10-s + (3.88 − 2.24i)11-s + (−3.48 + 6.02i)12-s + (−3.28 − 1.48i)13-s + 1.13i·15-s + (−2.48 + 4.29i)16-s + (−1.64 − 2.84i)17-s + (0.416 − 0.240i)18-s + (−4.52 − 2.60i)19-s + ⋯
L(s)  = 1  + (1.51 + 0.877i)2-s + (0.483 + 0.837i)3-s + (1.03 + 1.79i)4-s + (0.261 + 0.150i)5-s + 1.69i·6-s + 1.89i·8-s + (0.0323 − 0.0559i)9-s + (0.264 + 0.458i)10-s + (1.17 − 0.675i)11-s + (−1.00 + 1.74i)12-s + (−0.911 − 0.410i)13-s + 0.292i·15-s + (−0.620 + 1.07i)16-s + (−0.398 − 0.690i)17-s + (0.0982 − 0.0567i)18-s + (−1.03 − 0.598i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.346 - 0.938i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.346 - 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.346 - 0.938i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (324, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.346 - 0.938i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.32163 + 3.33225i\)
\(L(\frac12)\) \(\approx\) \(2.32163 + 3.33225i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 + (3.28 + 1.48i)T \)
good2 \( 1 + (-2.14 - 1.24i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (-0.837 - 1.45i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.584 - 0.337i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-3.88 + 2.24i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.64 + 2.84i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.52 + 2.60i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.38 - 4.12i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 9.31T + 29T^{2} \)
31 \( 1 + (1.41 - 0.818i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.25 - 0.721i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 7.92iT - 41T^{2} \)
43 \( 1 + 4.61T + 43T^{2} \)
47 \( 1 + (-6.81 - 3.93i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.57 - 2.73i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.20 + 1.27i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.15 - 2.00i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.36 + 3.67i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 7.75iT - 71T^{2} \)
73 \( 1 + (-13.1 + 7.57i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.33 - 12.6i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 1.45iT - 83T^{2} \)
89 \( 1 + (6.74 + 3.89i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 17.9iT - 97T^{2} \)
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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

−11.08576080011414621933939166648, −9.792668346679206369491458718988, −9.129868054882926332394949389537, −8.011537702290871767305151035442, −6.96181582115746031386873336595, −6.24963152219848466488139347535, −5.26704341476881601480551381306, −4.28490812130415208964798585150, −3.66471095735834991294326833129, −2.57836240386344460708876322764, 1.92721499000846287380879753288, 2.03402643849682407353968976116, 3.75695739782625370587056190221, 4.43054648453965804063577410181, 5.60864155118800419756385285845, 6.57464809123794571718628452701, 7.32717831146264665302072377197, 8.629584754110677444412595399888, 9.697111746862246246377215450802, 10.56721454959682649199119144364

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