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Properties

Label 2-252-12.11-c1-0-10
Degree $2$
Conductor $252$
Sign $0.236 + 0.971i$
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  + (1.16 − 0.795i)2-s + (0.735 − 1.85i)4-s − 0.665i·5-s − i·7-s + (−0.619 − 2.75i)8-s + (−0.529 − 0.778i)10-s − 2.07·11-s + 5.55·13-s + (−0.795 − 1.16i)14-s + (−2.91 − 2.73i)16-s + 2.16i·17-s + 4.49i·19-s + (−1.23 − 0.489i)20-s + (−2.43 + 1.65i)22-s − 4.28·23-s + ⋯
L(s)  = 1  + (0.826 − 0.562i)2-s + (0.367 − 0.929i)4-s − 0.297i·5-s − 0.377i·7-s + (−0.218 − 0.975i)8-s + (−0.167 − 0.246i)10-s − 0.627·11-s + 1.54·13-s + (−0.212 − 0.312i)14-s + (−0.729 − 0.683i)16-s + 0.524i·17-s + 1.03i·19-s + (−0.276 − 0.109i)20-s + (−0.518 + 0.352i)22-s − 0.892·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.236 + 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.236 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.236 + 0.971i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ 0.236 + 0.971i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.51115 - 1.18719i\)
\(L(\frac12)\) \(\approx\) \(1.51115 - 1.18719i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.16 + 0.795i)T \)
3 \( 1 \)
7 \( 1 + iT \)
good5 \( 1 + 0.665iT - 5T^{2} \)
11 \( 1 + 2.07T + 11T^{2} \)
13 \( 1 - 5.55T + 13T^{2} \)
17 \( 1 - 2.16iT - 17T^{2} \)
19 \( 1 - 4.49iT - 19T^{2} \)
23 \( 1 + 4.28T + 23T^{2} \)
29 \( 1 + 1.41iT - 29T^{2} \)
31 \( 1 - 6.61iT - 31T^{2} \)
37 \( 1 + 5.43T + 37T^{2} \)
41 \( 1 - 5.69iT - 41T^{2} \)
43 \( 1 + 2.11iT - 43T^{2} \)
47 \( 1 - 10.5T + 47T^{2} \)
53 \( 1 - 10.6iT - 53T^{2} \)
59 \( 1 + 13.5T + 59T^{2} \)
61 \( 1 + 0.615T + 61T^{2} \)
67 \( 1 + 14.0iT - 67T^{2} \)
71 \( 1 - 15.5T + 71T^{2} \)
73 \( 1 + 11.9T + 73T^{2} \)
79 \( 1 + 0.824iT - 79T^{2} \)
83 \( 1 - 6.36T + 83T^{2} \)
89 \( 1 + 12.6iT - 89T^{2} \)
97 \( 1 - 0.824T + 97T^{2} \)
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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

−12.05271821556009961610896717096, −10.73695304963290243780818494710, −10.45951145271319698628915091358, −9.051251817049801990026331290320, −7.926368050813395945334039369772, −6.47709646302430290869372219971, −5.59867949078185557889539244691, −4.32447074479859258734638117959, −3.29124727648994752648947847560, −1.49222272730507784093306498558, 2.56221559711809152169697217991, 3.81848354088664604816488037603, 5.15070037803000944357537272807, 6.11997805820152569297884061435, 7.09776298441615507110757287249, 8.206045672419147886907314772479, 9.080502820518962176522382551252, 10.66674717331599650881963140685, 11.39623575034901352994049271222, 12.40606983998455307919234582803

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