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Properties

Label 2-350-5.4-c5-0-5
Degree $2$
Conductor $350$
Sign $0.447 - 0.894i$
Analytic cond. $56.1343$
Root an. cond. $7.49228$
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  + 4i·2-s − 30.5i·3-s − 16·4-s + 122.·6-s − 49i·7-s − 64i·8-s − 688.·9-s + 392.·11-s + 488. i·12-s + 631. i·13-s + 196·14-s + 256·16-s − 1.37e3i·17-s − 2.75e3i·18-s − 1.49e3·19-s + ⋯
L(s)  = 1  + 0.707i·2-s − 1.95i·3-s − 0.5·4-s + 1.38·6-s − 0.377i·7-s − 0.353i·8-s − 2.83·9-s + 0.977·11-s + 0.978i·12-s + 1.03i·13-s + 0.267·14-s + 0.250·16-s − 1.15i·17-s − 2.00i·18-s − 0.951·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $0.447 - 0.894i$
Analytic conductor: \(56.1343\)
Root analytic conductor: \(7.49228\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{350} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :5/2),\ 0.447 - 0.894i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.8025866194\)
\(L(\frac12)\) \(\approx\) \(0.8025866194\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 4iT \)
5 \( 1 \)
7 \( 1 + 49iT \)
good3 \( 1 + 30.5iT - 243T^{2} \)
11 \( 1 - 392.T + 1.61e5T^{2} \)
13 \( 1 - 631. iT - 3.71e5T^{2} \)
17 \( 1 + 1.37e3iT - 1.41e6T^{2} \)
19 \( 1 + 1.49e3T + 2.47e6T^{2} \)
23 \( 1 - 4.57e3iT - 6.43e6T^{2} \)
29 \( 1 + 2.70e3T + 2.05e7T^{2} \)
31 \( 1 + 6.93e3T + 2.86e7T^{2} \)
37 \( 1 - 1.47e3iT - 6.93e7T^{2} \)
41 \( 1 - 1.47e3T + 1.15e8T^{2} \)
43 \( 1 - 1.07e4iT - 1.47e8T^{2} \)
47 \( 1 + 6.47e3iT - 2.29e8T^{2} \)
53 \( 1 + 3.26e3iT - 4.18e8T^{2} \)
59 \( 1 - 2.92e4T + 7.14e8T^{2} \)
61 \( 1 - 3.64e4T + 8.44e8T^{2} \)
67 \( 1 + 828. iT - 1.35e9T^{2} \)
71 \( 1 - 2.80e4T + 1.80e9T^{2} \)
73 \( 1 - 7.61e4iT - 2.07e9T^{2} \)
79 \( 1 + 1.07e4T + 3.07e9T^{2} \)
83 \( 1 + 9.40e4iT - 3.93e9T^{2} \)
89 \( 1 - 4.35e4T + 5.58e9T^{2} \)
97 \( 1 - 3.41e4iT - 8.58e9T^{2} \)
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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

−11.36643742148251685439886779579, −9.440416966792007375264390636997, −8.731595940885389872115164538467, −7.60793472131199628171980277959, −7.04184223399755267392231652464, −6.39380842962294787202257320497, −5.34764847568626433534420757804, −3.69797883337391125816662354975, −2.05410001086205215029834011087, −1.05555474268206049894674869583, 0.22983239363143018143963558278, 2.33695878649934662924640133673, 3.58518100012768525807928308158, 4.18135769023654329170920729020, 5.27630323059252197348574169161, 6.18847043161505100473618900970, 8.358675668637472495116638971210, 8.852656148892811130877915081459, 9.750316598884825227661974465408, 10.63038558236662043244008678136

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