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Properties

Label 2-1260-105.23-c1-0-12
Degree $2$
Conductor $1260$
Sign $-0.703 + 0.710i$
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  + (−1.11 − 1.93i)5-s + (−0.988 + 2.45i)7-s + (2.72 − 1.57i)11-s + (−0.380 − 0.380i)13-s + (−0.321 + 1.20i)17-s + (−5.59 − 3.22i)19-s + (−2.12 − 7.93i)23-s + (−2.49 + 4.33i)25-s + 4.09·29-s + (2.84 + 4.93i)31-s + (5.85 − 0.834i)35-s + (−2.89 − 10.7i)37-s + 7.39i·41-s + (−8.31 − 8.31i)43-s + (1.56 − 0.420i)47-s + ⋯
L(s)  = 1  + (−0.500 − 0.865i)5-s + (−0.373 + 0.927i)7-s + (0.820 − 0.473i)11-s + (−0.105 − 0.105i)13-s + (−0.0780 + 0.291i)17-s + (−1.28 − 0.740i)19-s + (−0.443 − 1.65i)23-s + (−0.498 + 0.866i)25-s + 0.761·29-s + (0.511 + 0.885i)31-s + (0.990 − 0.141i)35-s + (−0.475 − 1.77i)37-s + 1.15i·41-s + (−1.26 − 1.26i)43-s + (0.228 − 0.0612i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.703 + 0.710i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.703 + 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1260\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.703 + 0.710i$
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: \((2,\ 1260,\ (\ :1/2),\ -0.703 + 0.710i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7192166653\)
\(L(\frac12)\) \(\approx\) \(0.7192166653\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.11 + 1.93i)T \)
7 \( 1 + (0.988 - 2.45i)T \)
good11 \( 1 + (-2.72 + 1.57i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.380 + 0.380i)T + 13iT^{2} \)
17 \( 1 + (0.321 - 1.20i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (5.59 + 3.22i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.12 + 7.93i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 - 4.09T + 29T^{2} \)
31 \( 1 + (-2.84 - 4.93i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.89 + 10.7i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 7.39iT - 41T^{2} \)
43 \( 1 + (8.31 + 8.31i)T + 43iT^{2} \)
47 \( 1 + (-1.56 + 0.420i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (10.7 + 2.88i)T + (45.8 + 26.5i)T^{2} \)
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 \( 1 + (4.89 + 1.31i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 15.1iT - 71T^{2} \)
73 \( 1 + (2.51 - 9.39i)T + (-63.2 - 36.5i)T^{2} \)
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

−8.999845685092363519227831801833, −8.758360285752263054564300626844, −8.046478805195726020611254854458, −6.68566873246520498621061428497, −6.17340529930623390446501930567, −5.01538595980369646479514581626, −4.29077529580635778220361889148, −3.17994293848999236015208541682, −1.93120738762581935174912822163, −0.29839397773469261489961979755, 1.59258964276122199195719726281, 3.05619428430278365074638365975, 3.90667403618615307989401741868, 4.61004846198396696311376068893, 6.19279104945772343912910387503, 6.65177252873996711343287798312, 7.52633664475732419265260339211, 8.148261385030556182590266196919, 9.370805388783230397704978268431, 10.07945208336430681341311163136

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