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Properties

Label 2-170-85.63-c1-0-4
Degree $2$
Conductor $170$
Sign $0.754 - 0.656i$
Analytic cond. $1.35745$
Root an. cond. $1.16509$
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  + (0.923 + 0.382i)2-s + (0.0811 + 0.0161i)3-s + (0.707 + 0.707i)4-s + (1.07 + 1.96i)5-s + (0.0688 + 0.0459i)6-s + (0.143 − 0.214i)7-s + (0.382 + 0.923i)8-s + (−2.76 − 1.14i)9-s + (0.240 + 2.22i)10-s + (0.818 − 1.22i)11-s + (0.0459 + 0.0688i)12-s − 0.161i·13-s + (0.214 − 0.143i)14-s + (0.0553 + 0.176i)15-s + i·16-s + (4.08 − 0.561i)17-s + ⋯
L(s)  = 1  + (0.653 + 0.270i)2-s + (0.0468 + 0.00932i)3-s + (0.353 + 0.353i)4-s + (0.479 + 0.877i)5-s + (0.0280 + 0.0187i)6-s + (0.0541 − 0.0810i)7-s + (0.135 + 0.326i)8-s + (−0.921 − 0.381i)9-s + (0.0759 + 0.703i)10-s + (0.246 − 0.369i)11-s + (0.0132 + 0.0198i)12-s − 0.0448i·13-s + (0.0573 − 0.0382i)14-s + (0.0143 + 0.0456i)15-s + 0.250i·16-s + (0.990 − 0.136i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.754 - 0.656i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.754 - 0.656i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $0.754 - 0.656i$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{170} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 170,\ (\ :1/2),\ 0.754 - 0.656i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.57039 + 0.588109i\)
\(L(\frac12)\) \(\approx\) \(1.57039 + 0.588109i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.923 - 0.382i)T \)
5 \( 1 + (-1.07 - 1.96i)T \)
17 \( 1 + (-4.08 + 0.561i)T \)
good3 \( 1 + (-0.0811 - 0.0161i)T + (2.77 + 1.14i)T^{2} \)
7 \( 1 + (-0.143 + 0.214i)T + (-2.67 - 6.46i)T^{2} \)
11 \( 1 + (-0.818 + 1.22i)T + (-4.20 - 10.1i)T^{2} \)
13 \( 1 + 0.161iT - 13T^{2} \)
19 \( 1 + (1.05 - 0.436i)T + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (1.45 + 7.32i)T + (-21.2 + 8.80i)T^{2} \)
29 \( 1 + (5.25 + 1.04i)T + (26.7 + 11.0i)T^{2} \)
31 \( 1 + (4.63 + 6.94i)T + (-11.8 + 28.6i)T^{2} \)
37 \( 1 + (1.08 - 5.47i)T + (-34.1 - 14.1i)T^{2} \)
41 \( 1 + (5.60 - 1.11i)T + (37.8 - 15.6i)T^{2} \)
43 \( 1 + (-1.92 + 0.798i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 - 9.14T + 47T^{2} \)
53 \( 1 + (-0.350 + 0.845i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (-0.837 + 2.02i)T + (-41.7 - 41.7i)T^{2} \)
61 \( 1 + (-0.778 - 3.91i)T + (-56.3 + 23.3i)T^{2} \)
67 \( 1 + (-8.09 - 8.09i)T + 67iT^{2} \)
71 \( 1 + (8.03 - 5.36i)T + (27.1 - 65.5i)T^{2} \)
73 \( 1 + (-1.72 - 2.58i)T + (-27.9 + 67.4i)T^{2} \)
79 \( 1 + (0.595 + 0.398i)T + (30.2 + 72.9i)T^{2} \)
83 \( 1 + (-2.66 - 1.10i)T + (58.6 + 58.6i)T^{2} \)
89 \( 1 + (9.97 + 9.97i)T + 89iT^{2} \)
97 \( 1 + (1.95 + 2.93i)T + (-37.1 + 89.6i)T^{2} \)
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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

−13.03398561125957398890096673319, −11.89267578175362794338725952946, −11.07091632680718191277091425365, −10.01207075477229272403439262529, −8.719995640178670499532041978174, −7.49730741372512207875660219250, −6.30694149435017943742716419121, −5.58179527268828560091948486381, −3.80568486617153211652679354691, −2.59821780960171449969443868224, 1.86049610085552615679872933456, 3.61563774823300741128883955328, 5.17220309507120005612337045854, 5.78253885460971068363643655080, 7.41675749926132543230900173441, 8.708563918699751308042904124857, 9.650861939133060357279267136556, 10.82976966977332231385188914061, 11.91031060327337250396807268261, 12.61935039298411868153637265897

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