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Properties

Label 2-175-5.4-c5-0-12
Degree $2$
Conductor $175$
Sign $0.894 + 0.447i$
Analytic cond. $28.0671$
Root an. cond. $5.29784$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 8i·2-s − i·3-s − 32·4-s − 8·6-s + 49i·7-s + 242·9-s − 453·11-s + 32i·12-s + 969i·13-s + 392·14-s − 1.02e3·16-s + 1.63e3i·17-s − 1.93e3i·18-s + 1.55e3·19-s + 49·21-s + 3.62e3i·22-s + ⋯
L(s)  = 1  − 1.41i·2-s − 0.0641i·3-s − 4-s − 0.0907·6-s + 0.377i·7-s + 0.995·9-s − 1.12·11-s + 0.0641i·12-s + 1.59i·13-s + 0.534·14-s − 16-s + 1.37i·17-s − 1.40i·18-s + 0.985·19-s + 0.0242·21-s + 1.59i·22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $0.894 + 0.447i$
Analytic conductor: \(28.0671\)
Root analytic conductor: \(5.29784\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{175} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 175,\ (\ :5/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.672358744\)
\(L(\frac12)\) \(\approx\) \(1.672358744\)
\(L(\frac{7}{2})\) 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 \( 1 - 49iT \)
good2 \( 1 + 8iT - 32T^{2} \)
3 \( 1 + iT - 243T^{2} \)
11 \( 1 + 453T + 1.61e5T^{2} \)
13 \( 1 - 969iT - 3.71e5T^{2} \)
17 \( 1 - 1.63e3iT - 1.41e6T^{2} \)
19 \( 1 - 1.55e3T + 2.47e6T^{2} \)
23 \( 1 - 1.65e3iT - 6.43e6T^{2} \)
29 \( 1 - 4.98e3T + 2.05e7T^{2} \)
31 \( 1 - 1.19e3T + 2.86e7T^{2} \)
37 \( 1 + 1.10e4iT - 6.93e7T^{2} \)
41 \( 1 + 1.72e3T + 1.15e8T^{2} \)
43 \( 1 - 1.08e4iT - 1.47e8T^{2} \)
47 \( 1 - 2.62e4iT - 2.29e8T^{2} \)
53 \( 1 + 2.59e4iT - 4.18e8T^{2} \)
59 \( 1 - 4.58e3T + 7.14e8T^{2} \)
61 \( 1 + 1.24e4T + 8.44e8T^{2} \)
67 \( 1 + 1.58e4iT - 1.35e9T^{2} \)
71 \( 1 - 5.17e4T + 1.80e9T^{2} \)
73 \( 1 + 4.84e3iT - 2.07e9T^{2} \)
79 \( 1 + 6.27e4T + 3.07e9T^{2} \)
83 \( 1 - 2.36e4iT - 3.93e9T^{2} \)
89 \( 1 - 1.47e5T + 5.58e9T^{2} \)
97 \( 1 + 8.34e3iT - 8.58e9T^{2} \)
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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

−11.73568312719317848069050645179, −10.81416398259059334432302750730, −9.949361732436961955761367947815, −9.151686362148485108103715834549, −7.74151209681761902256821820158, −6.45815906732561642241387220150, −4.79492160768369488594983183889, −3.70362858934648110183946782235, −2.30001645617406793007499947497, −1.29807473905808848987781302870, 0.58060314810936465986601744244, 2.87743572919821925751961825977, 4.75251036030282506388032019195, 5.46025930056446819366695991473, 6.87149161634215483327583783636, 7.58292236807515149963801734584, 8.381313126003868411913195068632, 9.847386884642949521396264582937, 10.59753810391452638004688101755, 12.06735326588730341561040450758

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