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Properties

Label 2-175-35.34-c4-0-29
Degree $2$
Conductor $175$
Sign $0.487 + 0.873i$
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  + 1.17i·2-s − 9.10·3-s + 14.6·4-s − 10.6i·6-s + (27.5 − 40.5i)7-s + 35.9i·8-s + 1.87·9-s + 106.·11-s − 133.·12-s − 243.·13-s + (47.5 + 32.3i)14-s + 191.·16-s − 148.·17-s + 2.20i·18-s − 244. i·19-s + ⋯
L(s)  = 1  + 0.293i·2-s − 1.01·3-s + 0.913·4-s − 0.296i·6-s + (0.562 − 0.826i)7-s + 0.561i·8-s + 0.0231·9-s + 0.882·11-s − 0.924·12-s − 1.43·13-s + (0.242 + 0.165i)14-s + 0.749·16-s − 0.512·17-s + 0.00680i·18-s − 0.677i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.487 + 0.873i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.487 + 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $0.487 + 0.873i$
Analytic conductor: \(18.0897\)
Root analytic conductor: \(4.25320\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{175} (174, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 175,\ (\ :2),\ 0.487 + 0.873i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.381072955\)
\(L(\frac12)\) \(\approx\) \(1.381072955\)
\(L(3)\) 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 \( 1 + (-27.5 + 40.5i)T \)
good2 \( 1 - 1.17iT - 16T^{2} \)
3 \( 1 + 9.10T + 81T^{2} \)
11 \( 1 - 106.T + 1.46e4T^{2} \)
13 \( 1 + 243.T + 2.85e4T^{2} \)
17 \( 1 + 148.T + 8.35e4T^{2} \)
19 \( 1 + 244. iT - 1.30e5T^{2} \)
23 \( 1 + 621. iT - 2.79e5T^{2} \)
29 \( 1 - 774.T + 7.07e5T^{2} \)
31 \( 1 + 403. iT - 9.23e5T^{2} \)
37 \( 1 + 1.29e3iT - 1.87e6T^{2} \)
41 \( 1 + 2.53e3iT - 2.82e6T^{2} \)
43 \( 1 + 2.24e3iT - 3.41e6T^{2} \)
47 \( 1 - 2.49e3T + 4.87e6T^{2} \)
53 \( 1 - 281. iT - 7.89e6T^{2} \)
59 \( 1 + 5.82e3iT - 1.21e7T^{2} \)
61 \( 1 - 6.26e3iT - 1.38e7T^{2} \)
67 \( 1 - 4.12e3iT - 2.01e7T^{2} \)
71 \( 1 + 793.T + 2.54e7T^{2} \)
73 \( 1 - 1.60e3T + 2.83e7T^{2} \)
79 \( 1 + 9.85e3T + 3.89e7T^{2} \)
83 \( 1 - 617.T + 4.74e7T^{2} \)
89 \( 1 - 7.34e3iT - 6.27e7T^{2} \)
97 \( 1 + 67.6T + 8.85e7T^{2} \)
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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

−11.79720903044571181948250462473, −10.95741435694157867878041904831, −10.27087984225842054560452115746, −8.685202627974426501203974310220, −7.26309562064088400730761948468, −6.72988194702177387939606806146, −5.50292166294232882655994757993, −4.38297256230058798986423488798, −2.35804867222830132775757741387, −0.58923490452120119073890156438, 1.42840700008421445301506045757, 2.83227014427786595377096936132, 4.74265575839161879288107760832, 5.86478771306209056694312278277, 6.73005531941551871859544359322, 7.964480362602433905188606667014, 9.407476169484104768659349919717, 10.46678410641369144271098164945, 11.66223804419050195320660578834, 11.73832839745323278333283181597

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