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Properties

Label 2-175-35.19-c4-0-21
Degree $2$
Conductor $175$
Sign $0.647 + 0.762i$
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  + (−5.10 − 2.94i)2-s + (4.33 + 7.51i)3-s + (9.35 + 16.2i)4-s − 51.1i·6-s + (45.7 − 17.5i)7-s − 16.0i·8-s + (2.85 − 4.95i)9-s + (−37.1 − 64.3i)11-s + (−81.1 + 140. i)12-s − 28.3·13-s + (−285. − 44.9i)14-s + (102. − 177. i)16-s + (110. + 191. i)17-s + (−29.1 + 16.8i)18-s + (−515. − 297. i)19-s + ⋯
L(s)  = 1  + (−1.27 − 0.736i)2-s + (0.482 + 0.834i)3-s + (0.584 + 1.01i)4-s − 1.42i·6-s + (0.933 − 0.358i)7-s − 0.250i·8-s + (0.0352 − 0.0611i)9-s + (−0.306 − 0.531i)11-s + (−0.563 + 0.976i)12-s − 0.167·13-s + (−1.45 − 0.229i)14-s + (0.400 − 0.694i)16-s + (0.383 + 0.663i)17-s + (−0.0900 + 0.0519i)18-s + (−1.42 − 0.824i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.647 + 0.762i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.647 + 0.762i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $0.647 + 0.762i$
Analytic conductor: \(18.0897\)
Root analytic conductor: \(4.25320\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{175} (124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 175,\ (\ :2),\ 0.647 + 0.762i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.182308707\)
\(L(\frac12)\) \(\approx\) \(1.182308707\)
\(L(3)\) 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 \( 1 + (-45.7 + 17.5i)T \)
good2 \( 1 + (5.10 + 2.94i)T + (8 + 13.8i)T^{2} \)
3 \( 1 + (-4.33 - 7.51i)T + (-40.5 + 70.1i)T^{2} \)
11 \( 1 + (37.1 + 64.3i)T + (-7.32e3 + 1.26e4i)T^{2} \)
13 \( 1 + 28.3T + 2.85e4T^{2} \)
17 \( 1 + (-110. - 191. i)T + (-4.17e4 + 7.23e4i)T^{2} \)
19 \( 1 + (515. + 297. i)T + (6.51e4 + 1.12e5i)T^{2} \)
23 \( 1 + (-412. - 238. i)T + (1.39e5 + 2.42e5i)T^{2} \)
29 \( 1 - 1.26e3T + 7.07e5T^{2} \)
31 \( 1 + (432. - 249. i)T + (4.61e5 - 7.99e5i)T^{2} \)
37 \( 1 + (-80.0 - 46.2i)T + (9.37e5 + 1.62e6i)T^{2} \)
41 \( 1 + 1.75e3iT - 2.82e6T^{2} \)
43 \( 1 + 2.78e3iT - 3.41e6T^{2} \)
47 \( 1 + (-671. + 1.16e3i)T + (-2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + (-1.62e3 + 939. i)T + (3.94e6 - 6.83e6i)T^{2} \)
59 \( 1 + (-4.23e3 + 2.44e3i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (-3.30e3 - 1.90e3i)T + (6.92e6 + 1.19e7i)T^{2} \)
67 \( 1 + (3.68e3 - 2.12e3i)T + (1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 - 9.75e3T + 2.54e7T^{2} \)
73 \( 1 + (-234. - 406. i)T + (-1.41e7 + 2.45e7i)T^{2} \)
79 \( 1 + (5.39e3 - 9.34e3i)T + (-1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 - 7.15e3T + 4.74e7T^{2} \)
89 \( 1 + (1.00e4 + 5.82e3i)T + (3.13e7 + 5.43e7i)T^{2} \)
97 \( 1 - 6.00e3T + 8.85e7T^{2} \)
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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

−11.39217039752402688127350288240, −10.59022741019738732098888973233, −10.06663184951151702992651156382, −8.688079179944833312035807418134, −8.538729651352085967778799409708, −7.09011141571662802070969531333, −5.12700015580371941061449490833, −3.77593271992664478082262468044, −2.33063223730751801268343875622, −0.78550796040141268894391148532, 1.12711498949269939328948911829, 2.35869849094742342340962880200, 4.72862728276423380623986835034, 6.33465828787504656840890902819, 7.35929086652052203612147164552, 8.074119341522370124793621423746, 8.694621003012686884071155418504, 9.921169110598648365803489108774, 10.87157602088067339006475646377, 12.27087607877616463383680742088

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