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Properties

Label 2-612-204.203-c1-0-20
Degree $2$
Conductor $612$
Sign $0.403 + 0.915i$
Analytic cond. $4.88684$
Root an. cond. $2.21062$
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  − 1.41i·2-s − 2.00·4-s + 4.24·5-s + 2.82i·8-s − 6i·10-s + 4·13-s + 4.00·16-s + (−2.12 + 3.53i)17-s − 8.48·20-s + 12.9·25-s − 5.65i·26-s + 4.24·29-s − 5.65i·32-s + (5.00 + 3i)34-s − 12i·37-s + ⋯
L(s)  = 1  − 0.999i·2-s − 1.00·4-s + 1.89·5-s + 1.00i·8-s − 1.89i·10-s + 1.10·13-s + 1.00·16-s + (−0.514 + 0.857i)17-s − 1.89·20-s + 2.59·25-s − 1.10i·26-s + 0.787·29-s − 1.00i·32-s + (0.857 + 0.514i)34-s − 1.97i·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 612 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.403 + 0.915i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 612 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.403 + 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(612\)    =    \(2^{2} \cdot 3^{2} \cdot 17\)
Sign: $0.403 + 0.915i$
Analytic conductor: \(4.88684\)
Root analytic conductor: \(2.21062\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{612} (611, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 612,\ (\ :1/2),\ 0.403 + 0.915i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.57413 - 1.02671i\)
\(L(\frac12)\) \(\approx\) \(1.57413 - 1.02671i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41iT \)
3 \( 1 \)
17 \( 1 + (2.12 - 3.53i)T \)
good5 \( 1 - 4.24T + 5T^{2} \)
7 \( 1 + 7T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 - 4T + 13T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 - 4.24T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 12iT - 37T^{2} \)
41 \( 1 + 12.7T + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 7.07iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 12iT - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + 6iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 18.3iT - 89T^{2} \)
97 \( 1 - 18iT - 97T^{2} \)
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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

−10.57800020521757063207459569445, −9.711449739754592838307808238067, −9.019792580697769266187733693614, −8.293362797966283743041912031050, −6.56012848394888563171478145990, −5.82406983635837267046016990018, −4.90448517018594930684171028443, −3.57136937362775505058143052958, −2.29737296873673019062319256402, −1.41782464593145812618402367898, 1.43465017352053930939310262577, 3.05861174260679874024368265805, 4.70269060891022292300479711789, 5.47309714159647814068770226520, 6.39578760802427181320237802216, 6.82735631565073382983028544291, 8.351374099554834379360659646890, 8.933911177544503426285143134084, 9.856707297964413275165864859537, 10.33036184573592275883695698322

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