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Properties

Label 2-252-21.5-c1-0-1
Degree $2$
Conductor $252$
Sign $0.932 + 0.360i$
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  + (1.22 − 2.12i)5-s + (0.5 + 2.59i)7-s + (3.67 − 2.12i)11-s − 1.73i·13-s + (−2.44 − 4.24i)17-s + (4.5 + 2.59i)19-s + (−0.499 − 0.866i)25-s + 8.48i·29-s + (−1.5 + 0.866i)31-s + (6.12 + 2.12i)35-s + (2.5 − 4.33i)37-s − 12.2·41-s − 11·43-s + (1.22 − 2.12i)47-s + (−6.5 + 2.59i)49-s + ⋯
L(s)  = 1  + (0.547 − 0.948i)5-s + (0.188 + 0.981i)7-s + (1.10 − 0.639i)11-s − 0.480i·13-s + (−0.594 − 1.02i)17-s + (1.03 + 0.596i)19-s + (−0.0999 − 0.173i)25-s + 1.57i·29-s + (−0.269 + 0.155i)31-s + (1.03 + 0.358i)35-s + (0.410 − 0.711i)37-s − 1.91·41-s − 1.67·43-s + (0.178 − 0.309i)47-s + (−0.928 + 0.371i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.932 + 0.360i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.932 + 0.360i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.932 + 0.360i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ 0.932 + 0.360i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.39741 - 0.260314i\)
\(L(\frac12)\) \(\approx\) \(1.39741 - 0.260314i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 + (-0.5 - 2.59i)T \)
good5 \( 1 + (-1.22 + 2.12i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-3.67 + 2.12i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 1.73iT - 13T^{2} \)
17 \( 1 + (2.44 + 4.24i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.5 - 2.59i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 8.48iT - 29T^{2} \)
31 \( 1 + (1.5 - 0.866i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.5 + 4.33i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 12.2T + 41T^{2} \)
43 \( 1 + 11T + 43T^{2} \)
47 \( 1 + (-1.22 + 2.12i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (7.34 - 4.24i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.44 - 4.24i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3 + 1.73i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.5 - 6.06i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 12.7iT - 71T^{2} \)
73 \( 1 + (-13.5 + 7.79i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.5 - 9.52i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 12.2T + 83T^{2} \)
89 \( 1 + (-7.34 + 12.7i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 3.46iT - 97T^{2} \)
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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

−11.99509730254681749594520892533, −11.27576760193984546392324055542, −9.810973488816037138521372531002, −9.019580103677894110219882042715, −8.436413443175717186009511848225, −6.91630955251243010655675039666, −5.65361615539146904128759010102, −4.98688067689919651731722617299, −3.25574812476184287349372974983, −1.48893953575493608715093514627, 1.81632771225251732995576875751, 3.53388503834659934706710823264, 4.68206038359731570134866659811, 6.43287194056498319643870936649, 6.85266765662240481903303805564, 8.090514353022859697083395214164, 9.502991287470778359145247104636, 10.13102039273429221246788656354, 11.14307084075667396920288878304, 11.87782417582941681221237354312

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