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Properties

Label 2-1260-105.23-c1-0-14
Degree $2$
Conductor $1260$
Sign $-0.940 + 0.340i$
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  + (1.38 − 1.75i)5-s + (−2.64 + 0.0650i)7-s + (−1.79 + 1.03i)11-s + (−0.920 − 0.920i)13-s + (−0.416 + 1.55i)17-s + (−1.78 − 1.03i)19-s + (−1.31 − 4.89i)23-s + (−1.18 − 4.85i)25-s − 1.58·29-s + (−4.97 − 8.61i)31-s + (−3.53 + 4.74i)35-s + (1.21 + 4.52i)37-s + 11.8i·41-s + (−6.62 − 6.62i)43-s + (−9.91 + 2.65i)47-s + ⋯
L(s)  = 1  + (0.617 − 0.786i)5-s + (−0.999 + 0.0245i)7-s + (−0.540 + 0.312i)11-s + (−0.255 − 0.255i)13-s + (−0.100 + 0.376i)17-s + (−0.409 − 0.236i)19-s + (−0.273 − 1.01i)23-s + (−0.237 − 0.971i)25-s − 0.294·29-s + (−0.893 − 1.54i)31-s + (−0.598 + 0.801i)35-s + (0.199 + 0.744i)37-s + 1.84i·41-s + (−1.00 − 1.00i)43-s + (−1.44 + 0.387i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.940 + 0.340i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.940 + 0.340i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1260\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.940 + 0.340i$
Analytic conductor: \(10.0611\)
Root analytic conductor: \(3.17193\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1260} (233, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1260,\ (\ :1/2),\ -0.940 + 0.340i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5540945406\)
\(L(\frac12)\) \(\approx\) \(0.5540945406\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.38 + 1.75i)T \)
7 \( 1 + (2.64 - 0.0650i)T \)
good11 \( 1 + (1.79 - 1.03i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.920 + 0.920i)T + 13iT^{2} \)
17 \( 1 + (0.416 - 1.55i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (1.78 + 1.03i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.31 + 4.89i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + 1.58T + 29T^{2} \)
31 \( 1 + (4.97 + 8.61i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.21 - 4.52i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 11.8iT - 41T^{2} \)
43 \( 1 + (6.62 + 6.62i)T + 43iT^{2} \)
47 \( 1 + (9.91 - 2.65i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (5.81 + 1.55i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-4.46 - 7.72i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.41 + 7.65i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.45 + 1.99i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 1.25iT - 71T^{2} \)
73 \( 1 + (-2.24 + 8.38i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (2.95 + 1.70i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.626 + 0.626i)T - 83iT^{2} \)
89 \( 1 + (-2.79 + 4.83i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (8.71 - 8.71i)T - 97iT^{2} \)
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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

−9.522741193398973499431141896242, −8.544047525105605891593540572926, −7.84055530835541552535762088619, −6.64528814240890086005532208448, −6.04540019920639924901967529894, −5.08651290714510675949622034914, −4.22805690200586083038458260661, −2.95700323086904724242991635450, −1.90969629080524683233962392535, −0.21198634655718667518311335099, 1.88934348048085164142516399075, 2.99479001685635395322453410651, 3.71839355582909842065368685591, 5.20494136807192910324148120600, 5.90898854052780539407366034503, 6.82579601517182583563305172305, 7.33672637059533253042064355458, 8.529688428993007560508061383310, 9.443161492072356622742267801462, 9.974918581564694155192800406142

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