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Properties

Label 2-175-5.2-c4-0-33
Degree $2$
Conductor $175$
Sign $-0.973 + 0.229i$
Analytic cond. $18.0897$
Root an. cond. $4.25320$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.896 − 0.896i)2-s + (9.60 − 9.60i)3-s − 14.3i·4-s − 17.2·6-s + (13.0 + 13.0i)7-s + (−27.2 + 27.2i)8-s − 103. i·9-s − 106.·11-s + (−138. − 138. i)12-s + (222. − 222. i)13-s − 23.4i·14-s − 181.·16-s + (−163. − 163. i)17-s + (−92.8 + 92.8i)18-s − 406. i·19-s + ⋯
L(s)  = 1  + (−0.224 − 0.224i)2-s + (1.06 − 1.06i)3-s − 0.899i·4-s − 0.478·6-s + (0.267 + 0.267i)7-s + (−0.425 + 0.425i)8-s − 1.27i·9-s − 0.879·11-s + (−0.959 − 0.959i)12-s + (1.31 − 1.31i)13-s − 0.119i·14-s − 0.708·16-s + (−0.567 − 0.567i)17-s + (−0.286 + 0.286i)18-s − 1.12i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 + 0.229i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $-0.973 + 0.229i$
Analytic conductor: \(18.0897\)
Root analytic conductor: \(4.25320\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{175} (57, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 175,\ (\ :2),\ -0.973 + 0.229i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.003270426\)
\(L(\frac12)\) \(\approx\) \(2.003270426\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 + (-13.0 - 13.0i)T \)
good2 \( 1 + (0.896 + 0.896i)T + 16iT^{2} \)
3 \( 1 + (-9.60 + 9.60i)T - 81iT^{2} \)
11 \( 1 + 106.T + 1.46e4T^{2} \)
13 \( 1 + (-222. + 222. i)T - 2.85e4iT^{2} \)
17 \( 1 + (163. + 163. i)T + 8.35e4iT^{2} \)
19 \( 1 + 406. iT - 1.30e5T^{2} \)
23 \( 1 + (492. - 492. i)T - 2.79e5iT^{2} \)
29 \( 1 - 371. iT - 7.07e5T^{2} \)
31 \( 1 + 315.T + 9.23e5T^{2} \)
37 \( 1 + (-1.11e3 - 1.11e3i)T + 1.87e6iT^{2} \)
41 \( 1 - 273.T + 2.82e6T^{2} \)
43 \( 1 + (-739. + 739. i)T - 3.41e6iT^{2} \)
47 \( 1 + (326. + 326. i)T + 4.87e6iT^{2} \)
53 \( 1 + (502. - 502. i)T - 7.89e6iT^{2} \)
59 \( 1 + 5.70e3iT - 1.21e7T^{2} \)
61 \( 1 + 974.T + 1.38e7T^{2} \)
67 \( 1 + (-1.63e3 - 1.63e3i)T + 2.01e7iT^{2} \)
71 \( 1 - 586.T + 2.54e7T^{2} \)
73 \( 1 + (-7.09e3 + 7.09e3i)T - 2.83e7iT^{2} \)
79 \( 1 - 2.38e3iT - 3.89e7T^{2} \)
83 \( 1 + (-1.29e3 + 1.29e3i)T - 4.74e7iT^{2} \)
89 \( 1 + 1.14e4iT - 6.27e7T^{2} \)
97 \( 1 + (-6.91e3 - 6.91e3i)T + 8.85e7iT^{2} \)
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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

−11.46153234730759480891368164623, −10.61073068129432278715814087492, −9.354807883522690147517655657716, −8.465365023174999654454564944062, −7.65113800519906343514478523498, −6.34070498343489677939983386960, −5.19300332360083865178542075275, −3.06255736710726896634741181712, −1.98488375313911063945474408631, −0.69006092000875636980334577188, 2.30651815184102082219887689472, 3.77601911608083347440944682145, 4.28557093388628830054052354088, 6.27550024015865947109696099627, 7.80626281377205397517457268283, 8.452743637048284856252355521176, 9.224222607318667044147963002124, 10.33355847560193046906909096136, 11.30572433017268687876290385594, 12.62929167899509144711775344069

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