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Properties

Label 2-175-5.3-c4-0-24
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  + (4.43 − 4.43i)2-s + (10.8 + 10.8i)3-s − 23.3i·4-s + 95.9·6-s + (−13.0 + 13.0i)7-s + (−32.7 − 32.7i)8-s + 152. i·9-s + 208.·11-s + (252. − 252. i)12-s + (1.92 + 1.92i)13-s + 116. i·14-s + 83.2·16-s + (−164. + 164. i)17-s + (677. + 677. i)18-s − 212. i·19-s + ⋯
L(s)  = 1  + (1.10 − 1.10i)2-s + (1.20 + 1.20i)3-s − 1.46i·4-s + 2.66·6-s + (−0.267 + 0.267i)7-s + (−0.512 − 0.512i)8-s + 1.88i·9-s + 1.72·11-s + (1.75 − 1.75i)12-s + (0.0113 + 0.0113i)13-s + 0.593i·14-s + 0.325·16-s + (−0.570 + 0.570i)17-s + (2.09 + 2.09i)18-s − 0.587i·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: $\chi_{175} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 175,\ (\ :2),\ 0.973 + 0.229i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(5.113257320\)
\(L(\frac12)\) \(\approx\) \(5.113257320\)
\(L(3)\) 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 \( 1 + (-4.43 + 4.43i)T - 16iT^{2} \)
3 \( 1 + (-10.8 - 10.8i)T + 81iT^{2} \)
11 \( 1 - 208.T + 1.46e4T^{2} \)
13 \( 1 + (-1.92 - 1.92i)T + 2.85e4iT^{2} \)
17 \( 1 + (164. - 164. i)T - 8.35e4iT^{2} \)
19 \( 1 + 212. iT - 1.30e5T^{2} \)
23 \( 1 + (391. + 391. i)T + 2.79e5iT^{2} \)
29 \( 1 + 36.5iT - 7.07e5T^{2} \)
31 \( 1 - 349.T + 9.23e5T^{2} \)
37 \( 1 + (479. - 479. i)T - 1.87e6iT^{2} \)
41 \( 1 - 959.T + 2.82e6T^{2} \)
43 \( 1 + (1.35e3 + 1.35e3i)T + 3.41e6iT^{2} \)
47 \( 1 + (1.89e3 - 1.89e3i)T - 4.87e6iT^{2} \)
53 \( 1 + (2.83e3 + 2.83e3i)T + 7.89e6iT^{2} \)
59 \( 1 + 3.14e3iT - 1.21e7T^{2} \)
61 \( 1 + 1.73e3T + 1.38e7T^{2} \)
67 \( 1 + (-245. + 245. i)T - 2.01e7iT^{2} \)
71 \( 1 + 6.00e3T + 2.54e7T^{2} \)
73 \( 1 + (-2.33e3 - 2.33e3i)T + 2.83e7iT^{2} \)
79 \( 1 + 6.60e3iT - 3.89e7T^{2} \)
83 \( 1 + (5.77e3 + 5.77e3i)T + 4.74e7iT^{2} \)
89 \( 1 - 1.07e4iT - 6.27e7T^{2} \)
97 \( 1 + (3.20e3 - 3.20e3i)T - 8.85e7iT^{2} \)
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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

−11.99550678497281894947040238568, −11.06897238505323769848400361963, −10.07530553816493496327741909371, −9.272139093379202257835758321417, −8.352146107584205175006911594551, −6.36030815182530400640643905196, −4.72937538897901368015430586041, −3.98811470895479941317318930466, −3.13078242891109928798720776889, −1.90497934650916334739823526583, 1.47540378640428839899954307321, 3.24424130475206138254302075468, 4.22579771446498187425517486767, 6.06863676754913839906740301182, 6.81611859783804543873653785982, 7.55612125213916695205649070525, 8.581063069710676804992980775318, 9.628019084416509760521291468396, 11.73228540921337284255583922106, 12.53027596461374960640985991912

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