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Properties

Label 2-1260-105.53-c1-0-7
Degree $2$
Conductor $1260$
Sign $0.990 + 0.140i$
Analytic cond. $10.0611$
Root an. cond. $3.17193$
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  + (2.23 − 0.00183i)5-s + (−2.45 + 0.988i)7-s + (−2.72 − 1.57i)11-s + (−0.380 − 0.380i)13-s + (1.20 − 0.321i)17-s + (5.59 − 3.22i)19-s + (7.93 + 2.12i)23-s + (4.99 − 0.00820i)25-s + 4.09·29-s + (2.84 − 4.93i)31-s + (−5.48 + 2.21i)35-s + (10.7 + 2.89i)37-s + 7.39i·41-s + (−8.31 − 8.31i)43-s + (−0.420 + 1.56i)47-s + ⋯
L(s)  = 1  + (0.999 − 0.000820i)5-s + (−0.927 + 0.373i)7-s + (−0.820 − 0.473i)11-s + (−0.105 − 0.105i)13-s + (0.291 − 0.0780i)17-s + (1.28 − 0.740i)19-s + (1.65 + 0.443i)23-s + (0.999 − 0.00164i)25-s + 0.761·29-s + (0.511 − 0.885i)31-s + (−0.927 + 0.374i)35-s + (1.77 + 0.475i)37-s + 1.15i·41-s + (−1.26 − 1.26i)43-s + (−0.0612 + 0.228i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.140i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 + 0.140i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1260\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.990 + 0.140i$
Analytic conductor: \(10.0611\)
Root analytic conductor: \(3.17193\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1260} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1260,\ (\ :1/2),\ 0.990 + 0.140i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.805287856\)
\(L(\frac12)\) \(\approx\) \(1.805287856\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (-2.23 + 0.00183i)T \)
7 \( 1 + (2.45 - 0.988i)T \)
good11 \( 1 + (2.72 + 1.57i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.380 + 0.380i)T + 13iT^{2} \)
17 \( 1 + (-1.20 + 0.321i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-5.59 + 3.22i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-7.93 - 2.12i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 - 4.09T + 29T^{2} \)
31 \( 1 + (-2.84 + 4.93i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-10.7 - 2.89i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 7.39iT - 41T^{2} \)
43 \( 1 + (8.31 + 8.31i)T + 43iT^{2} \)
47 \( 1 + (0.420 - 1.56i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-2.88 - 10.7i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (5.50 - 9.52i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.74 + 6.47i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.31 - 4.89i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 15.1iT - 71T^{2} \)
73 \( 1 + (-9.39 + 2.51i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (3.95 - 2.28i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.60 - 2.60i)T - 83iT^{2} \)
89 \( 1 + (-4.35 - 7.53i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.02 + 2.02i)T - 97iT^{2} \)
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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

−9.550167450922192978921409579276, −9.164574249751958405086292905959, −8.069674840794042411908927060954, −7.10601125213879682729227280451, −6.25935969490272547877619312710, −5.52374317021138179279023747649, −4.78626298377706523914032437087, −3.05746103617088487370846623253, −2.72848475926599521151117585800, −0.980090449991548071862244456724, 1.09525921905812743518527300396, 2.57285309464503677789769787176, 3.33492901780168742805434668827, 4.77366570922063427700700830282, 5.47489397682763099772177392020, 6.46381857780704895194793711491, 7.09337667784808872599349781396, 8.058774238588006025554974154076, 9.115015996826622610378074191405, 9.852698100287938138005382729092

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