Properties

Degree $2$
Conductor $170$
Sign $-0.447 - 0.894i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·2-s + i·3-s − 4-s + (1 + 2i)5-s − 6-s i·8-s + 2·9-s + (−2 + i)10-s − 6·11-s i·12-s + 3i·13-s + (−2 + i)15-s + 16-s i·17-s + 2i·18-s + 7·19-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.577i·3-s − 0.5·4-s + (0.447 + 0.894i)5-s − 0.408·6-s − 0.353i·8-s + 0.666·9-s + (−0.632 + 0.316i)10-s − 1.80·11-s − 0.288i·12-s + 0.832i·13-s + (−0.516 + 0.258i)15-s + 0.250·16-s − 0.242i·17-s + 0.471i·18-s + 1.60·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $-0.447 - 0.894i$
Motivic weight: \(1\)
Character: $\chi_{170} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 170,\ (\ :1/2),\ -0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.596833 + 0.965696i\)
\(L(\frac12)\) \(\approx\) \(0.596833 + 0.965696i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
5 \( 1 + (-1 - 2i)T \)
17 \( 1 + iT \)
good3 \( 1 - iT - 3T^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 + 6T + 11T^{2} \)
13 \( 1 - 3iT - 13T^{2} \)
19 \( 1 - 7T + 19T^{2} \)
23 \( 1 + 8iT - 23T^{2} \)
29 \( 1 - 5T + 29T^{2} \)
31 \( 1 - 5T + 31T^{2} \)
37 \( 1 + 8iT - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 + 3iT - 47T^{2} \)
53 \( 1 - 9iT - 53T^{2} \)
59 \( 1 + 5T + 59T^{2} \)
61 \( 1 + 3T + 61T^{2} \)
67 \( 1 - 2iT - 67T^{2} \)
71 \( 1 + 15T + 71T^{2} \)
73 \( 1 + 11iT - 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 - 4iT - 83T^{2} \)
89 \( 1 - T + 89T^{2} \)
97 \( 1 - 9iT - 97T^{2} \)
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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

−13.45019682525548528185759574713, −12.19723532688957565209501873946, −10.65836645624033867377691971627, −10.18243183921051514266936658515, −9.132870746615799306542864492406, −7.70620778082021457200679001867, −6.88711588675284328701798294168, −5.58041120481242809972795257783, −4.48216253547213087813980636822, −2.78849164707539717056843579898, 1.24620747171930182430763415435, 2.91003541838342100486717755651, 4.81233862370992299921138409531, 5.69149124039611196796630011913, 7.54852750955059439015496471914, 8.264859617764210124365604697928, 9.756618677118959182678570781437, 10.24808373329158249543853928376, 11.68846318239185755529641597631, 12.56902754620698119956140156272

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