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Properties

Label 2-18e2-9.7-c1-0-0
Degree $2$
Conductor $324$
Sign $0.173 - 0.984i$
Analytic cond. $2.58715$
Root an. cond. $1.60846$
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  + (−2.5 + 4.33i)7-s + (3.5 + 6.06i)13-s − 19-s + (2.5 − 4.33i)25-s + (2 + 3.46i)31-s − 37-s + (−4 + 6.92i)43-s + (−9.00 − 15.5i)49-s + (6.5 − 11.2i)61-s + (−5.5 − 9.52i)67-s + 17·73-s + (6.5 − 11.2i)79-s − 35·91-s + (−2.5 + 4.33i)97-s + (3.5 + 6.06i)103-s + ⋯
L(s)  = 1  + (−0.944 + 1.63i)7-s + (0.970 + 1.68i)13-s − 0.229·19-s + (0.5 − 0.866i)25-s + (0.359 + 0.622i)31-s − 0.164·37-s + (−0.609 + 1.05i)43-s + (−1.28 − 2.22i)49-s + (0.832 − 1.44i)61-s + (−0.671 − 1.16i)67-s + 1.98·73-s + (0.731 − 1.26i)79-s − 3.66·91-s + (−0.253 + 0.439i)97-s + (0.344 + 0.597i)103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $0.173 - 0.984i$
Analytic conductor: \(2.58715\)
Root analytic conductor: \(1.60846\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (217, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :1/2),\ 0.173 - 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.852540 + 0.715366i\)
\(L(\frac12)\) \(\approx\) \(0.852540 + 0.715366i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5 \( 1 + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (2.5 - 4.33i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.5 - 6.06i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + T + 19T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2 - 3.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + T + 37T^{2} \)
41 \( 1 + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4 - 6.92i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.5 + 11.2i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.5 + 9.52i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 17T + 73T^{2} \)
79 \( 1 + (-6.5 + 11.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + (2.5 - 4.33i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.92893567873738895808040613308, −11.03604763223208315750936001424, −9.734468935517001051997820790788, −9.021788410912916157624860181979, −8.342785903395719378047634157708, −6.61546418584568748996332354964, −6.20200305017867850384522648625, −4.84678644916868500454409584126, −3.40833744756611229016433241825, −2.10046681190686105816304517361, 0.825202739181669871320106696033, 3.17112831268645506320487135411, 4.01074531449814063409273287254, 5.52542362782316614216500455019, 6.64253610396501901993766338709, 7.50780876952033811786354442138, 8.499816978974123648859951075744, 9.800061902910841349221896659816, 10.46900672684705922151516453328, 11.12611800332127043043445309373

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