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Properties

Label 2-1260-105.2-c1-0-7
Degree $2$
Conductor $1260$
Sign $0.259 - 0.965i$
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  + (1.14 + 1.91i)5-s + (0.642 + 2.56i)7-s + (2.01 − 1.16i)11-s + (0.104 − 0.104i)13-s + (3.64 + 0.977i)17-s + (−0.646 − 0.373i)19-s + (−1.75 + 0.471i)23-s + (−2.37 + 4.40i)25-s + 7.38·29-s + (0.668 + 1.15i)31-s + (−4.19 + 4.17i)35-s + (−6.66 + 1.78i)37-s − 0.0508i·41-s + (−3.06 + 3.06i)43-s + (−0.746 − 2.78i)47-s + ⋯
L(s)  = 1  + (0.512 + 0.858i)5-s + (0.242 + 0.970i)7-s + (0.608 − 0.351i)11-s + (0.0289 − 0.0289i)13-s + (0.884 + 0.237i)17-s + (−0.148 − 0.0855i)19-s + (−0.366 + 0.0982i)23-s + (−0.474 + 0.880i)25-s + 1.37·29-s + (0.120 + 0.207i)31-s + (−0.708 + 0.705i)35-s + (−1.09 + 0.293i)37-s − 0.00794i·41-s + (−0.466 + 0.466i)43-s + (−0.108 − 0.406i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.259 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.259 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1260\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.259 - 0.965i$
Analytic conductor: \(10.0611\)
Root analytic conductor: \(3.17193\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1260} (737, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1260,\ (\ :1/2),\ 0.259 - 0.965i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.886361526\)
\(L(\frac12)\) \(\approx\) \(1.886361526\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.14 - 1.91i)T \)
7 \( 1 + (-0.642 - 2.56i)T \)
good11 \( 1 + (-2.01 + 1.16i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.104 + 0.104i)T - 13iT^{2} \)
17 \( 1 + (-3.64 - 0.977i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (0.646 + 0.373i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.75 - 0.471i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 7.38T + 29T^{2} \)
31 \( 1 + (-0.668 - 1.15i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (6.66 - 1.78i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 0.0508iT - 41T^{2} \)
43 \( 1 + (3.06 - 3.06i)T - 43iT^{2} \)
47 \( 1 + (0.746 + 2.78i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (0.248 - 0.926i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-3.37 - 5.83i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.61 - 6.25i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.62 - 9.80i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 12.8iT - 71T^{2} \)
73 \( 1 + (-10.9 - 2.94i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (9.97 + 5.75i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.461 + 0.461i)T + 83iT^{2} \)
89 \( 1 + (-2.21 + 3.83i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.18 - 5.18i)T + 97iT^{2} \)
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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

−9.984573212596237024866985147136, −8.992477465747398745947162674336, −8.367111180918252984136189018753, −7.34489065153569594582457171220, −6.38606455775029316604528229701, −5.85217748354744344150888441770, −4.88723599905737613048938713218, −3.53660993396302169448812062175, −2.69999420199489241660730104717, −1.55269245892348981710132955738, 0.861927074764382066671901683378, 1.91127401346962972640462465394, 3.47905732147217570244527732257, 4.44678486069065553881318326953, 5.15095665086919394396961522763, 6.21847741216002102223053605035, 7.03721635749029601538058820026, 7.996454129472163803076212602328, 8.661445470827627029448329019153, 9.690184497861766841969599449947

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