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Properties

Label 2-175-5.4-c3-0-25
Degree $2$
Conductor $175$
Sign $0.894 - 0.447i$
Analytic cond. $10.3253$
Root an. cond. $3.21330$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5.41i·2-s − 4.65i·3-s − 21.3·4-s − 25.2·6-s + 7i·7-s + 72.0i·8-s + 5.31·9-s − 52.2·11-s + 99.2i·12-s + 30.6i·13-s + 37.8·14-s + 219.·16-s − 37.2i·17-s − 28.7i·18-s − 80.2·19-s + ⋯
L(s)  = 1  − 1.91i·2-s − 0.896i·3-s − 2.66·4-s − 1.71·6-s + 0.377i·7-s + 3.18i·8-s + 0.196·9-s − 1.43·11-s + 2.38i·12-s + 0.654i·13-s + 0.723·14-s + 3.43·16-s − 0.531i·17-s − 0.376i·18-s − 0.968·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $0.894 - 0.447i$
Analytic conductor: \(10.3253\)
Root analytic conductor: \(3.21330\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{175} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 175,\ (\ :3/2),\ 0.894 - 0.447i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.179880 + 0.0424641i\)
\(L(\frac12)\) \(\approx\) \(0.179880 + 0.0424641i\)
\(L(\frac{5}{2})\) 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 \( 1 - 7iT \)
good2 \( 1 + 5.41iT - 8T^{2} \)
3 \( 1 + 4.65iT - 27T^{2} \)
11 \( 1 + 52.2T + 1.33e3T^{2} \)
13 \( 1 - 30.6iT - 2.19e3T^{2} \)
17 \( 1 + 37.2iT - 4.91e3T^{2} \)
19 \( 1 + 80.2T + 6.85e3T^{2} \)
23 \( 1 - 25.8iT - 1.21e4T^{2} \)
29 \( 1 + 20.9T + 2.43e4T^{2} \)
31 \( 1 + 314.T + 2.97e4T^{2} \)
37 \( 1 + 197. iT - 5.06e4T^{2} \)
41 \( 1 - 11.3T + 6.89e4T^{2} \)
43 \( 1 + 33.8iT - 7.95e4T^{2} \)
47 \( 1 - 361. iT - 1.03e5T^{2} \)
53 \( 1 - 153. iT - 1.48e5T^{2} \)
59 \( 1 - 616T + 2.05e5T^{2} \)
61 \( 1 - 15.2T + 2.26e5T^{2} \)
67 \( 1 - 166. iT - 3.00e5T^{2} \)
71 \( 1 + 952T + 3.57e5T^{2} \)
73 \( 1 + 148. iT - 3.89e5T^{2} \)
79 \( 1 + 857.T + 4.93e5T^{2} \)
83 \( 1 - 660. iT - 5.71e5T^{2} \)
89 \( 1 - 45.7T + 7.04e5T^{2} \)
97 \( 1 + 1.68e3iT - 9.12e5T^{2} \)
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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

−11.38757766780310008942033801105, −10.60296608980760606100641000970, −9.569119792558090229642504129007, −8.566241642016171058844163407854, −7.40419676289149172650288038104, −5.52710301275790288475786828140, −4.23483860846907235139540127449, −2.66115853286060713733204180563, −1.74516262062128442405345671298, −0.083734173178597046867991202572, 3.77159704298825971939449878680, 4.85389613113856134963068168195, 5.67687892204942882487254397911, 7.00198318539079095395752752894, 7.933834704143804790678720291842, 8.815919869524161113562222466758, 10.02421380149923641707570459736, 10.61014808331374396470194695711, 12.90501000446557701785023609528, 13.23501176466808942655379794623

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