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Properties

Label 2-174-87.17-c1-0-2
Degree $2$
Conductor $174$
Sign $0.0164 - 0.999i$
Analytic cond. $1.38939$
Root an. cond. $1.17872$
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  + (0.707 + 0.707i)2-s + (1.10 + 1.33i)3-s + 1.00i·4-s − 3.14·5-s + (−0.158 + 1.72i)6-s + 2·7-s + (−0.707 + 0.707i)8-s + (−0.548 + 2.94i)9-s + (−2.22 − 2.22i)10-s + (3.85 + 3.85i)11-s + (−1.33 + 1.10i)12-s − 5.44i·13-s + (1.41 + 1.41i)14-s + (−3.48 − 4.19i)15-s − 1.00·16-s + (−3.14 − 3.14i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (0.639 + 0.769i)3-s + 0.500i·4-s − 1.40·5-s + (−0.0648 + 0.704i)6-s + 0.755·7-s + (−0.250 + 0.250i)8-s + (−0.182 + 0.983i)9-s + (−0.703 − 0.703i)10-s + (1.16 + 1.16i)11-s + (−0.384 + 0.319i)12-s − 1.51i·13-s + (0.377 + 0.377i)14-s + (−0.899 − 1.08i)15-s − 0.250·16-s + (−0.763 − 0.763i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 174 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0164 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 174 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0164 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(174\)    =    \(2 \cdot 3 \cdot 29\)
Sign: $0.0164 - 0.999i$
Analytic conductor: \(1.38939\)
Root analytic conductor: \(1.17872\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{174} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 174,\ (\ :1/2),\ 0.0164 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10811 + 1.09008i\)
\(L(\frac12)\) \(\approx\) \(1.10811 + 1.09008i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
3 \( 1 + (-1.10 - 1.33i)T \)
29 \( 1 + (-5.19 + 1.41i)T \)
good5 \( 1 + 3.14T + 5T^{2} \)
7 \( 1 - 2T + 7T^{2} \)
11 \( 1 + (-3.85 - 3.85i)T + 11iT^{2} \)
13 \( 1 + 5.44iT - 13T^{2} \)
17 \( 1 + (3.14 + 3.14i)T + 17iT^{2} \)
19 \( 1 + (-2.44 + 2.44i)T - 19iT^{2} \)
23 \( 1 + 4.87iT - 23T^{2} \)
31 \( 1 + (0.775 - 0.775i)T - 31iT^{2} \)
37 \( 1 + (-4 - 4i)T + 37iT^{2} \)
41 \( 1 + (5.51 - 5.51i)T - 41iT^{2} \)
43 \( 1 + (1.77 - 1.77i)T - 43iT^{2} \)
47 \( 1 + (-0.707 + 0.707i)T - 47iT^{2} \)
53 \( 1 + 8.16iT - 53T^{2} \)
59 \( 1 + 5.51iT - 59T^{2} \)
61 \( 1 + (5.89 - 5.89i)T - 61iT^{2} \)
67 \( 1 - 0.898iT - 67T^{2} \)
71 \( 1 + 8.34T + 71T^{2} \)
73 \( 1 + (-6 - 6i)T + 73iT^{2} \)
79 \( 1 + (11.0 - 11.0i)T - 79iT^{2} \)
83 \( 1 + 8.34iT - 83T^{2} \)
89 \( 1 + (-0.778 - 0.778i)T + 89iT^{2} \)
97 \( 1 + (7.34 + 7.34i)T + 97iT^{2} \)
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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

−13.06174992894413812305168867024, −11.91710916415909963668568720238, −11.22193498320437950314034837870, −9.902092214822152345823512575573, −8.586183202045646216601307199213, −7.912356665000389078214092962465, −6.92431232962470809899908145880, −4.88094486470147320794029800963, −4.34232359480972253612635393588, −3.01546259603006479369442447369, 1.53532904529844523546226607168, 3.49482639778846448970209304303, 4.26176775038323695327643981190, 6.21626288632165052587964494414, 7.32662840405766895620545782032, 8.429909847997759679536525775866, 9.153892971723022979259118832090, 11.06999933250035139656035838075, 11.72010381965439866469231476565, 12.17348803241271466520250449611

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