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Properties

Label 2-26e2-13.8-c2-0-22
Degree $2$
Conductor $676$
Sign $0.123 + 0.992i$
Analytic cond. $18.4196$
Root an. cond. $4.29181$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.46·3-s + (−5.87 − 5.87i)5-s + (3.11 − 3.11i)7-s + 20.9·9-s + (−3.04 + 3.04i)11-s + (−32.1 − 32.1i)15-s − 25.5i·17-s + (8.38 + 8.38i)19-s + (17.0 − 17.0i)21-s − 12.7i·23-s + 44.0i·25-s + 65.1·27-s − 10.7·29-s + (−16.1 − 16.1i)31-s + (−16.6 + 16.6i)33-s + ⋯
L(s)  = 1  + 1.82·3-s + (−1.17 − 1.17i)5-s + (0.445 − 0.445i)7-s + 2.32·9-s + (−0.276 + 0.276i)11-s + (−2.14 − 2.14i)15-s − 1.50i·17-s + (0.441 + 0.441i)19-s + (0.811 − 0.811i)21-s − 0.554i·23-s + 1.76i·25-s + 2.41·27-s − 0.371·29-s + (−0.522 − 0.522i)31-s + (−0.504 + 0.504i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.123 + 0.992i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.123 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(676\)    =    \(2^{2} \cdot 13^{2}\)
Sign: $0.123 + 0.992i$
Analytic conductor: \(18.4196\)
Root analytic conductor: \(4.29181\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{676} (437, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 676,\ (\ :1),\ 0.123 + 0.992i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.843123441\)
\(L(\frac12)\) \(\approx\) \(2.843123441\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 \)
good3 \( 1 - 5.46T + 9T^{2} \)
5 \( 1 + (5.87 + 5.87i)T + 25iT^{2} \)
7 \( 1 + (-3.11 + 3.11i)T - 49iT^{2} \)
11 \( 1 + (3.04 - 3.04i)T - 121iT^{2} \)
17 \( 1 + 25.5iT - 289T^{2} \)
19 \( 1 + (-8.38 - 8.38i)T + 361iT^{2} \)
23 \( 1 + 12.7iT - 529T^{2} \)
29 \( 1 + 10.7T + 841T^{2} \)
31 \( 1 + (16.1 + 16.1i)T + 961iT^{2} \)
37 \( 1 + (-37.3 + 37.3i)T - 1.36e3iT^{2} \)
41 \( 1 + (24.2 + 24.2i)T + 1.68e3iT^{2} \)
43 \( 1 + 20.7iT - 1.84e3T^{2} \)
47 \( 1 + (-22.6 + 22.6i)T - 2.20e3iT^{2} \)
53 \( 1 + 41.5T + 2.80e3T^{2} \)
59 \( 1 + (-23.3 + 23.3i)T - 3.48e3iT^{2} \)
61 \( 1 + 74.2T + 3.72e3T^{2} \)
67 \( 1 + (-46.1 - 46.1i)T + 4.48e3iT^{2} \)
71 \( 1 + (-30.8 - 30.8i)T + 5.04e3iT^{2} \)
73 \( 1 + (80.1 - 80.1i)T - 5.32e3iT^{2} \)
79 \( 1 - 144.T + 6.24e3T^{2} \)
83 \( 1 + (-116. - 116. i)T + 6.88e3iT^{2} \)
89 \( 1 + (6.47 - 6.47i)T - 7.92e3iT^{2} \)
97 \( 1 + (-52.9 - 52.9i)T + 9.40e3iT^{2} \)
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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

−9.654372012074111603869482133761, −9.118689002523845721951253126087, −8.293203902387832834042003950359, −7.67560147374790845168107409504, −7.21656102927669673503940800126, −5.10759134264972610997285215821, −4.27316605555942954580746817001, −3.55859806593823289239291700433, −2.27854838200597695199916076053, −0.847059845296881570475760548363, 1.84282981849024631949662555950, 3.05052273812041119082641269781, 3.51439787398582701936853143664, 4.58490654487387521088590440587, 6.33148754603709363123941896885, 7.41911493579524442790292831612, 7.925527894433397176392126133302, 8.522587350731285248206389323483, 9.468224646380210875497942148883, 10.47247631840082828666678742883

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