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Properties

Label 2-1260-35.4-c1-0-14
Degree $2$
Conductor $1260$
Sign $0.154 + 0.987i$
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  + (1.63 + 1.52i)5-s + (−2.63 − 0.209i)7-s + (1.13 − 1.97i)11-s − 6.09i·13-s + (−4.13 − 2.38i)17-s + (−2.13 − 3.70i)19-s + (−0.774 + 0.447i)23-s + (0.362 + 4.98i)25-s − 3.27·29-s + (2.13 − 3.70i)31-s + (−4 − 4.35i)35-s + (4.86 − 2.80i)37-s + 11.2·41-s − 6.50i·43-s + (1.86 − 1.07i)47-s + ⋯
L(s)  = 1  + (0.732 + 0.680i)5-s + (−0.996 − 0.0791i)7-s + (0.342 − 0.594i)11-s − 1.68i·13-s + (−1.00 − 0.579i)17-s + (−0.490 − 0.849i)19-s + (−0.161 + 0.0932i)23-s + (0.0725 + 0.997i)25-s − 0.608·29-s + (0.383 − 0.664i)31-s + (−0.676 − 0.736i)35-s + (0.799 − 0.461i)37-s + 1.76·41-s − 0.992i·43-s + (0.271 − 0.156i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.154 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.154 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1260\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.154 + 0.987i$
Analytic conductor: \(10.0611\)
Root analytic conductor: \(3.17193\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1260} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1260,\ (\ :1/2),\ 0.154 + 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.267694489\)
\(L(\frac12)\) \(\approx\) \(1.267694489\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.63 - 1.52i)T \)
7 \( 1 + (2.63 + 0.209i)T \)
good11 \( 1 + (-1.13 + 1.97i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 6.09iT - 13T^{2} \)
17 \( 1 + (4.13 + 2.38i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.13 + 3.70i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.774 - 0.447i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 3.27T + 29T^{2} \)
31 \( 1 + (-2.13 + 3.70i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.86 + 2.80i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 11.2T + 41T^{2} \)
43 \( 1 + 6.50iT - 43T^{2} \)
47 \( 1 + (-1.86 + 1.07i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.41 - 3.70i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.13 - 3.70i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.774 + 1.34i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (12.0 + 6.95i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 10.5T + 71T^{2} \)
73 \( 1 + (1.86 + 1.07i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.137 + 0.238i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 5.67iT - 83T^{2} \)
89 \( 1 + (-3.5 - 6.06i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 6.92iT - 97T^{2} \)
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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

−9.392447691936521757672521837991, −9.006967816043154187609391611037, −7.73004627609224625507685586972, −6.96904795191260782240398688951, −6.07759716339303186940954427498, −5.62718944979534829878674511504, −4.21250104063995072317681592691, −3.06794143683729159550521373678, −2.46077636247035058170246733541, −0.52167308500442181722868983472, 1.53660409079139769810263213102, 2.47117564713415783575172636537, 4.04949981252214918072348622374, 4.54902825005995032602449651792, 5.98001184783345053321431001008, 6.36744414287966394319601080208, 7.25958157780102463384805735129, 8.542345592675692784202428595196, 9.161995197016762435432223599960, 9.688135737931805657621570978185

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