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Properties

Label 2-13e2-13.2-c2-0-5
Degree $2$
Conductor $169$
Sign $0.466 - 0.884i$
Analytic cond. $4.60491$
Root an. cond. $2.14590$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.20 − 0.858i)2-s + (1.5 + 2.59i)3-s + (6.06 + 3.5i)4-s + (−2.34 + 2.34i)5-s + (−2.57 − 9.61i)6-s + (9.61 − 2.57i)7-s + (−7.03 − 7.03i)8-s + (9.52 − 5.5i)10-s + (−1.71 + 6.40i)11-s + 21i·12-s − 33·14-s + (−9.61 − 2.57i)15-s + (2.49 + 4.33i)16-s + (2.59 + 1.5i)17-s + (5.15 + 19.2i)19-s + (−22.4 + 6.00i)20-s + ⋯
L(s)  = 1  + (−1.60 − 0.429i)2-s + (0.5 + 0.866i)3-s + (1.51 + 0.875i)4-s + (−0.469 + 0.469i)5-s + (−0.429 − 1.60i)6-s + (1.37 − 0.367i)7-s + (−0.879 − 0.879i)8-s + (0.952 − 0.550i)10-s + (−0.156 + 0.582i)11-s + 1.75i·12-s − 2.35·14-s + (−0.640 − 0.171i)15-s + (0.156 + 0.270i)16-s + (0.152 + 0.0882i)17-s + (0.271 + 1.01i)19-s + (−1.12 + 0.300i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.466 - 0.884i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.466 - 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $0.466 - 0.884i$
Analytic conductor: \(4.60491\)
Root analytic conductor: \(2.14590\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (80, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 169,\ (\ :1),\ 0.466 - 0.884i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.718296 + 0.433410i\)
\(L(\frac12)\) \(\approx\) \(0.718296 + 0.433410i\)
\(L(2)\) 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 \( 1 + (3.20 + 0.858i)T + (3.46 + 2i)T^{2} \)
3 \( 1 + (-1.5 - 2.59i)T + (-4.5 + 7.79i)T^{2} \)
5 \( 1 + (2.34 - 2.34i)T - 25iT^{2} \)
7 \( 1 + (-9.61 + 2.57i)T + (42.4 - 24.5i)T^{2} \)
11 \( 1 + (1.71 - 6.40i)T + (-104. - 60.5i)T^{2} \)
17 \( 1 + (-2.59 - 1.5i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-5.15 - 19.2i)T + (-312. + 180.5i)T^{2} \)
23 \( 1 + (10.3 - 6i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-21 - 36.3i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (-28.1 + 28.1i)T - 961iT^{2} \)
37 \( 1 + (12.8 - 48.0i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 + (44.8 + 12.0i)T + (1.45e3 + 840.5i)T^{2} \)
43 \( 1 + (42.4 + 24.5i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-2.34 - 2.34i)T + 2.20e3iT^{2} \)
53 \( 1 + 24T + 2.80e3T^{2} \)
59 \( 1 + (-51.2 + 13.7i)T + (3.01e3 - 1.74e3i)T^{2} \)
61 \( 1 + (15 - 25.9i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-38.4 - 10.3i)T + (3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + (6.00 + 22.4i)T + (-4.36e3 + 2.52e3i)T^{2} \)
73 \( 1 + (28.1 + 28.1i)T + 5.32e3iT^{2} \)
79 \( 1 - 54T + 6.24e3T^{2} \)
83 \( 1 + (32.8 - 32.8i)T - 6.88e3iT^{2} \)
89 \( 1 + (-42.9 + 160. i)T + (-6.85e3 - 3.96e3i)T^{2} \)
97 \( 1 + (-5.15 - 19.2i)T + (-8.14e3 + 4.70e3i)T^{2} \)
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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

−12.08809313583924104626077259694, −11.36549564586994729918974358352, −10.32640702401923004293301019143, −9.933786352359915190939843187195, −8.636798287074699358204306753318, −7.989811449608565091429681007804, −7.03998232128868701594060047431, −4.76571077294677210905156305182, −3.35414497872606038237715613860, −1.59271636956302383774188862051, 0.912020867882832893194658233092, 2.24640555134595149452501706415, 4.85643807956893440104572000467, 6.55557100448021722005814690858, 7.70238840865830561390739262683, 8.264698902656337500075462757401, 8.752026978253210029948766202367, 10.18366524353680606810043995815, 11.27057705638109144170511252943, 12.08158537243959159751942888048

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