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Properties

Label 2-1218-203.104-c1-0-2
Degree $2$
Conductor $1218$
Sign $0.774 - 0.632i$
Analytic cond. $9.72577$
Root an. cond. $3.11861$
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  + (0.707 − 0.707i)2-s + (−0.707 − 0.707i)3-s − 1.00i·4-s − 3.88·5-s − 1.00·6-s + (−0.687 − 2.55i)7-s + (−0.707 − 0.707i)8-s + 1.00i·9-s + (−2.74 + 2.74i)10-s + (−2.40 + 2.40i)11-s + (−0.707 + 0.707i)12-s − 1.76·13-s + (−2.29 − 1.32i)14-s + (2.74 + 2.74i)15-s − 1.00·16-s + (2.16 + 2.16i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (−0.408 − 0.408i)3-s − 0.500i·4-s − 1.73·5-s − 0.408·6-s + (−0.259 − 0.965i)7-s + (−0.250 − 0.250i)8-s + 0.333i·9-s + (−0.867 + 0.867i)10-s + (−0.724 + 0.724i)11-s + (−0.204 + 0.204i)12-s − 0.490·13-s + (−0.612 − 0.352i)14-s + (0.708 + 0.708i)15-s − 0.250·16-s + (0.525 + 0.525i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1218 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.774 - 0.632i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1218 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.774 - 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1218\)    =    \(2 \cdot 3 \cdot 7 \cdot 29\)
Sign: $0.774 - 0.632i$
Analytic conductor: \(9.72577\)
Root analytic conductor: \(3.11861\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1218} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1218,\ (\ :1/2),\ 0.774 - 0.632i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4875855919\)
\(L(\frac12)\) \(\approx\) \(0.4875855919\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 0.707i)T \)
3 \( 1 + (0.707 + 0.707i)T \)
7 \( 1 + (0.687 + 2.55i)T \)
29 \( 1 + (-3.09 - 4.40i)T \)
good5 \( 1 + 3.88T + 5T^{2} \)
11 \( 1 + (2.40 - 2.40i)T - 11iT^{2} \)
13 \( 1 + 1.76T + 13T^{2} \)
17 \( 1 + (-2.16 - 2.16i)T + 17iT^{2} \)
19 \( 1 + (1.23 + 1.23i)T + 19iT^{2} \)
23 \( 1 - 7.16T + 23T^{2} \)
31 \( 1 + (-0.785 - 0.785i)T + 31iT^{2} \)
37 \( 1 + (6.21 + 6.21i)T + 37iT^{2} \)
41 \( 1 + (0.510 - 0.510i)T - 41iT^{2} \)
43 \( 1 + (1.06 - 1.06i)T - 43iT^{2} \)
47 \( 1 + (-1.53 + 1.53i)T - 47iT^{2} \)
53 \( 1 + 4.74T + 53T^{2} \)
59 \( 1 - 9.85iT - 59T^{2} \)
61 \( 1 + (-3.88 - 3.88i)T + 61iT^{2} \)
67 \( 1 - 1.35iT - 67T^{2} \)
71 \( 1 - 11.6iT - 71T^{2} \)
73 \( 1 + (11.0 - 11.0i)T - 73iT^{2} \)
79 \( 1 + (-7.92 + 7.92i)T - 79iT^{2} \)
83 \( 1 + 4.45iT - 83T^{2} \)
89 \( 1 + (-5.59 - 5.59i)T + 89iT^{2} \)
97 \( 1 + (5.97 - 5.97i)T - 97iT^{2} \)
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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

−10.33704723062523212358343225993, −8.961689881679076127877664200095, −7.939901177355719389960004030240, −7.21910702818280365195252377229, −6.81533827179145488693677914431, −5.26934512799309751935708053389, −4.56605792522781272343623690185, −3.75206236365263428688395471509, −2.80100454690039321353905023290, −1.06208466630131574689041180012, 0.22503612170936070990601899936, 2.90954962421789700579151144821, 3.47431814045484292034500105737, 4.71177583749615123238740606369, 5.17669325522852129820811291618, 6.25995617824477051700322557277, 7.14888750705839169612013644274, 8.031688602795891167571873904921, 8.536491734764160825801890476121, 9.521408874858469151836256924017

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