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Properties

Label 2-252-28.19-c1-0-11
Degree $2$
Conductor $252$
Sign $0.0633 + 0.997i$
Analytic cond. $2.01223$
Root an. cond. $1.41853$
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  − 1.41i·2-s − 2.00·4-s + (2.44 + 1.41i)5-s + (−0.5 − 2.59i)7-s + 2.82i·8-s + (2.00 − 3.46i)10-s + (4.89 − 2.82i)11-s − 5.19i·13-s + (−3.67 + 0.707i)14-s + 4.00·16-s + (−2.44 + 1.41i)17-s + (−2.5 + 4.33i)19-s + (−4.89 − 2.82i)20-s + (−4.00 − 6.92i)22-s + (2.44 + 1.41i)23-s + ⋯
L(s)  = 1  − 0.999i·2-s − 1.00·4-s + (1.09 + 0.632i)5-s + (−0.188 − 0.981i)7-s + 1.00i·8-s + (0.632 − 1.09i)10-s + (1.47 − 0.852i)11-s − 1.44i·13-s + (−0.981 + 0.188i)14-s + 1.00·16-s + (−0.594 + 0.342i)17-s + (−0.573 + 0.993i)19-s + (−1.09 − 0.632i)20-s + (−0.852 − 1.47i)22-s + (0.510 + 0.294i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0633 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.0633 + 0.997i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ 0.0633 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.983963 - 0.923500i\)
\(L(\frac12)\) \(\approx\) \(0.983963 - 0.923500i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41iT \)
3 \( 1 \)
7 \( 1 + (0.5 + 2.59i)T \)
good5 \( 1 + (-2.44 - 1.41i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-4.89 + 2.82i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 5.19iT - 13T^{2} \)
17 \( 1 + (2.44 - 1.41i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.5 - 4.33i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.44 - 1.41i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.5 - 4.33i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 5.65iT - 41T^{2} \)
43 \( 1 + 5.19iT - 43T^{2} \)
47 \( 1 + (2.44 - 4.24i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.44 - 4.24i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.44 - 4.24i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6 - 3.46i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.5 - 4.33i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 2.82iT - 71T^{2} \)
73 \( 1 + (-1.5 + 0.866i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.5 - 0.866i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 14.6T + 83T^{2} \)
89 \( 1 + (-9.79 - 5.65i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−11.65414267572772403986079298765, −10.61323919421374717290742442920, −10.25578377243328459473742035213, −9.229616351084120998375736829898, −8.155914965078829730146522786306, −6.60485004324887634354065297097, −5.67895953639799342751335759086, −4.03965004642630839321048018092, −3.02117077382766834952601299097, −1.33188539154039416810237689520, 1.93681423428097278607356714809, 4.27315387244006221465383174028, 5.19708902530992899686553365156, 6.41522103010728652854079228786, 6.90383382993225781726015243010, 8.832628613838791894614099369428, 9.079848608668255944764467314310, 9.765634452285832337119462144432, 11.53033453306582291091748125297, 12.51638125916046102527040097625

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