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Properties

Label 2-13e2-13.12-c3-0-32
Degree $2$
Conductor $169$
Sign $-0.832 - 0.554i$
Analytic cond. $9.97132$
Root an. cond. $3.15774$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4i·2-s + 2·3-s − 8·4-s − 17i·5-s − 8i·6-s + 20i·7-s − 23·9-s − 68·10-s − 32i·11-s − 16·12-s + 80·14-s − 34i·15-s − 64·16-s + 13·17-s + 92i·18-s − 30i·19-s + ⋯
L(s)  = 1  − 1.41i·2-s + 0.384·3-s − 4-s − 1.52i·5-s − 0.544i·6-s + 1.07i·7-s − 0.851·9-s − 2.15·10-s − 0.877i·11-s − 0.384·12-s + 1.52·14-s − 0.585i·15-s − 16-s + 0.185·17-s + 1.20i·18-s − 0.362i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 - 0.554i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.832 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $-0.832 - 0.554i$
Analytic conductor: \(9.97132\)
Root analytic conductor: \(3.15774\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (168, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 169,\ (\ :3/2),\ -0.832 - 0.554i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.396451 + 1.30939i\)
\(L(\frac12)\) \(\approx\) \(0.396451 + 1.30939i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
good2 \( 1 + 4iT - 8T^{2} \)
3 \( 1 - 2T + 27T^{2} \)
5 \( 1 + 17iT - 125T^{2} \)
7 \( 1 - 20iT - 343T^{2} \)
11 \( 1 + 32iT - 1.33e3T^{2} \)
17 \( 1 - 13T + 4.91e3T^{2} \)
19 \( 1 + 30iT - 6.85e3T^{2} \)
23 \( 1 + 78T + 1.21e4T^{2} \)
29 \( 1 - 197T + 2.43e4T^{2} \)
31 \( 1 - 74iT - 2.97e4T^{2} \)
37 \( 1 + 227iT - 5.06e4T^{2} \)
41 \( 1 - 165iT - 6.89e4T^{2} \)
43 \( 1 - 156T + 7.95e4T^{2} \)
47 \( 1 + 162iT - 1.03e5T^{2} \)
53 \( 1 - 93T + 1.48e5T^{2} \)
59 \( 1 + 864iT - 2.05e5T^{2} \)
61 \( 1 - 145T + 2.26e5T^{2} \)
67 \( 1 + 862iT - 3.00e5T^{2} \)
71 \( 1 + 654iT - 3.57e5T^{2} \)
73 \( 1 - 215iT - 3.89e5T^{2} \)
79 \( 1 + 76T + 4.93e5T^{2} \)
83 \( 1 + 628iT - 5.71e5T^{2} \)
89 \( 1 + 266iT - 7.04e5T^{2} \)
97 \( 1 + 238iT - 9.12e5T^{2} \)
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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

−11.92537294894514125911929604550, −10.97624425685648443780676057681, −9.579865359672247773109724927646, −8.845837243218007598649097974825, −8.276401013218958246843191233252, −5.96285018364300971255046241847, −4.83797653620749561727780672107, −3.34975771531299576180588028786, −2.13530020034696233179555476573, −0.58393472699915772495894284526, 2.64845798473429096116732732845, 4.16642901205392821823134307539, 5.82263150630822782155823716768, 6.83000658517553618462641463910, 7.45620588077621232973791973388, 8.363613101597092188964043044616, 9.862713283273342184544010865529, 10.72420058292812594306114924541, 11.83007192361508393306535944457, 13.59607737309315724784886601890

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