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Properties

Label 2-637-91.24-c1-0-12
Degree $2$
Conductor $637$
Sign $0.0325 - 0.999i$
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  + (1.34 − 0.360i)2-s + 1.44i·3-s + (−0.0536 + 0.0309i)4-s + (0.643 + 0.172i)5-s + (0.521 + 1.94i)6-s + (−2.02 + 2.02i)8-s + 0.901·9-s + 0.926·10-s + (−3.40 + 3.40i)11-s + (−0.0448 − 0.0776i)12-s + (3.60 + 0.0282i)13-s + (−0.249 + 0.931i)15-s + (−1.93 + 3.35i)16-s + (−0.233 − 0.405i)17-s + (1.21 − 0.324i)18-s + (−2.38 + 2.38i)19-s + ⋯
L(s)  = 1  + (0.950 − 0.254i)2-s + 0.836i·3-s + (−0.0268 + 0.0154i)4-s + (0.287 + 0.0770i)5-s + (0.213 + 0.795i)6-s + (−0.717 + 0.717i)8-s + 0.300·9-s + 0.293·10-s + (−1.02 + 1.02i)11-s + (−0.0129 − 0.0224i)12-s + (0.999 + 0.00782i)13-s + (−0.0644 + 0.240i)15-s + (−0.484 + 0.838i)16-s + (−0.0567 − 0.0982i)17-s + (0.285 − 0.0765i)18-s + (−0.547 + 0.547i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0325 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0325 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.0325 - 0.999i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (570, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.0325 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.51680 + 1.46815i\)
\(L(\frac12)\) \(\approx\) \(1.51680 + 1.46815i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.60 - 0.0282i)T \)
good2 \( 1 + (-1.34 + 0.360i)T + (1.73 - i)T^{2} \)
3 \( 1 - 1.44iT - 3T^{2} \)
5 \( 1 + (-0.643 - 0.172i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (3.40 - 3.40i)T - 11iT^{2} \)
17 \( 1 + (0.233 + 0.405i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.38 - 2.38i)T - 19iT^{2} \)
23 \( 1 + (-6.02 - 3.47i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.01 + 3.49i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.09 - 4.10i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (0.873 + 3.26i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (-3.68 - 0.986i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-3.42 - 1.97i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.58 + 9.64i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-2.20 + 3.81i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.58 + 5.91i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 - 4.87iT - 61T^{2} \)
67 \( 1 + (-6.61 - 6.61i)T + 67iT^{2} \)
71 \( 1 + (-3.19 + 0.857i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (-0.112 + 0.0301i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-0.194 - 0.337i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-11.5 + 11.5i)T - 83iT^{2} \)
89 \( 1 + (9.34 - 2.50i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (4.61 + 17.2i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.77733992216804749860529413097, −10.05393762388209828994947627651, −9.234466930192554876657182671063, −8.288408079662896883514124876578, −7.16293721857070671706885705777, −5.86191537287928179283246355887, −5.08106872943871509302286201293, −4.25348056268855140860493749870, −3.45754037710084067347552754148, −2.14423155521449902304992008845, 0.891604784936852838227444268442, 2.61821783420454051905275048338, 3.82361004816164136752677464627, 4.95612785708187982477740586367, 5.89877186612103298393809875973, 6.49322922913414904178450375270, 7.53633674321891297913251898387, 8.542751795901355802138753854836, 9.368391654276599005175161492826, 10.58767580499436249494813969578

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