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Properties

Label 2-175-5.2-c4-0-23
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  + (2.88 + 2.88i)2-s + (−11.3 + 11.3i)3-s + 0.699i·4-s − 65.5·6-s + (−13.0 − 13.0i)7-s + (44.2 − 44.2i)8-s − 176. i·9-s − 15.6·11-s + (−7.93 − 7.93i)12-s + (137. − 137. i)13-s − 75.6i·14-s + 266.·16-s + (−108. − 108. i)17-s + (510. − 510. i)18-s + 375. i·19-s + ⋯
L(s)  = 1  + (0.722 + 0.722i)2-s + (−1.26 + 1.26i)3-s + 0.0437i·4-s − 1.82·6-s + (−0.267 − 0.267i)7-s + (0.690 − 0.690i)8-s − 2.18i·9-s − 0.129·11-s + (−0.0551 − 0.0551i)12-s + (0.815 − 0.815i)13-s − 0.386i·14-s + 1.04·16-s + (−0.376 − 0.376i)17-s + (1.57 − 1.57i)18-s + 1.03i·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

\(L(\frac{5}{2})\) \(\approx\) \(1.490975061\)
\(L(\frac12)\) \(\approx\) \(1.490975061\)
\(L(3)\) 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 \( 1 + (-2.88 - 2.88i)T + 16iT^{2} \)
3 \( 1 + (11.3 - 11.3i)T - 81iT^{2} \)
11 \( 1 + 15.6T + 1.46e4T^{2} \)
13 \( 1 + (-137. + 137. i)T - 2.85e4iT^{2} \)
17 \( 1 + (108. + 108. i)T + 8.35e4iT^{2} \)
19 \( 1 - 375. iT - 1.30e5T^{2} \)
23 \( 1 + (153. - 153. i)T - 2.79e5iT^{2} \)
29 \( 1 + 1.53e3iT - 7.07e5T^{2} \)
31 \( 1 + 201.T + 9.23e5T^{2} \)
37 \( 1 + (-186. - 186. i)T + 1.87e6iT^{2} \)
41 \( 1 - 3.01e3T + 2.82e6T^{2} \)
43 \( 1 + (-2.33e3 + 2.33e3i)T - 3.41e6iT^{2} \)
47 \( 1 + (-922. - 922. i)T + 4.87e6iT^{2} \)
53 \( 1 + (-700. + 700. i)T - 7.89e6iT^{2} \)
59 \( 1 + 760. iT - 1.21e7T^{2} \)
61 \( 1 + 2.82e3T + 1.38e7T^{2} \)
67 \( 1 + (2.58e3 + 2.58e3i)T + 2.01e7iT^{2} \)
71 \( 1 + 5.87e3T + 2.54e7T^{2} \)
73 \( 1 + (-3.61e3 + 3.61e3i)T - 2.83e7iT^{2} \)
79 \( 1 + 7.49e3iT - 3.89e7T^{2} \)
83 \( 1 + (-2.53e3 + 2.53e3i)T - 4.74e7iT^{2} \)
89 \( 1 + 2.54e3iT - 6.27e7T^{2} \)
97 \( 1 + (7.25e3 + 7.25e3i)T + 8.85e7iT^{2} \)
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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

−12.03630182840040472297069106344, −10.87436973669149373856375737017, −10.30270165711053956399703400833, −9.378353052691402393645879420105, −7.59992779118319140469076654499, −6.05141874761176795336028022579, −5.82671603729816801001474399223, −4.54213832310846180961491003669, −3.74658941168586660429538563606, −0.58602277780742504719686017412, 1.26028540236406925973950756040, 2.53331389151685988202196731661, 4.32838145532193896248940940749, 5.54193704983357415242584764516, 6.55919285290865563714925707352, 7.54182596960959707695396364416, 8.869358603285110779076998618796, 10.84890819506501215297750112216, 11.16070964013041395096374028422, 12.17247995346046275512077928832

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