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Properties

Label 2-1197-7.2-c1-0-21
Degree $2$
Conductor $1197$
Sign $0.605 - 0.795i$
Analytic cond. $9.55809$
Root an. cond. $3.09161$
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  + (−1.20 + 2.09i)2-s + (−1.91 − 3.31i)4-s + (−1.91 + 3.31i)5-s + (−2.5 + 0.866i)7-s + 4.41·8-s + (−4.62 − 8.00i)10-s + (−1.91 − 3.31i)11-s + 2.82·13-s + (1.20 − 6.27i)14-s + (−1.49 + 2.59i)16-s + (−1.82 − 3.16i)17-s + (0.5 − 0.866i)19-s + 14.6·20-s + 9.24·22-s + (−0.914 + 1.58i)23-s + ⋯
L(s)  = 1  + (−0.853 + 1.47i)2-s + (−0.957 − 1.65i)4-s + (−0.856 + 1.48i)5-s + (−0.944 + 0.327i)7-s + 1.56·8-s + (−1.46 − 2.53i)10-s + (−0.577 − 0.999i)11-s + 0.784·13-s + (0.322 − 1.67i)14-s + (−0.374 + 0.649i)16-s + (−0.443 − 0.768i)17-s + (0.114 − 0.198i)19-s + 3.27·20-s + 1.97·22-s + (−0.190 + 0.330i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1197 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1197 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1197\)    =    \(3^{2} \cdot 7 \cdot 19\)
Sign: $0.605 - 0.795i$
Analytic conductor: \(9.55809\)
Root analytic conductor: \(3.09161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1197} (856, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1197,\ (\ :1/2),\ 0.605 - 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3802803195\)
\(L(\frac12)\) \(\approx\) \(0.3802803195\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (2.5 - 0.866i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (1.20 - 2.09i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (1.91 - 3.31i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.91 + 3.31i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 2.82T + 13T^{2} \)
17 \( 1 + (1.82 + 3.16i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (0.914 - 1.58i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 4.82T + 29T^{2} \)
31 \( 1 + (-1.58 - 2.74i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.585 - 1.01i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 7.65T + 41T^{2} \)
43 \( 1 + 12.6T + 43T^{2} \)
47 \( 1 + (-3.08 + 5.34i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1 + 1.73i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.41 - 9.37i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.32 + 12.6i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6 - 10.3i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 4.34T + 71T^{2} \)
73 \( 1 + (-5.15 - 8.93i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.17 - 2.02i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 4.17T + 83T^{2} \)
89 \( 1 + (-5.41 + 9.37i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 16.8T + 97T^{2} \)
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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

−9.846588323822392261282494779628, −8.550424039715521486568395933682, −8.365507246535292898652055355058, −7.16752289837293546859392416953, −6.78242389420464245946818812207, −6.13265707886720857768484133400, −5.18753776104196125701391248354, −3.58821010623645382947533449766, −2.85540912777409563501934767219, −0.31881857102812892313305110498, 0.823986142426020843845012321275, 1.97683138588962601431209226804, 3.37715543550858345895631610487, 4.08889344020454856569104777149, 4.91954512135716712612353027577, 6.39911898033500299256491633855, 7.66347294132190768843180388578, 8.398247007132390861484565174982, 8.860526706448331049178883160646, 9.809404045943410626281343786925

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